TL;DR: In this scene Aragorn confirms to Pippin that their relationship remains one of liege-lord and vassal within the feudal system of Gondor and Arnor. This confers high status and honour upon Pippin, as well as obligation: in fact, these are two sides of the same relationship.
When Aragorn says that his “realm lies also in the north” he is referring to the kingdom of Arnor. The history as described during the Council of Elrond:
Of Númenor he [Elrond] spoke, its glory and its fall, and the return of the Kings of Men to Middle-earth out of the deeps of the Sea, borne upon the wings of storm. Then Elendil the Tall and his mighty sons, Isildur and Anárion, became great lords; and the North-realm they made in Arnor, and the South-realm in Gondor above the mouths of Anduin.
J. R. R. Tolkien (1954). The Fellowship of the Ring, book II, chapter 2. London: George Allen and Unwin.
After the War of the Last Alliance, Gondor flourished but Arnor did not:
‘In the North after the war and the slaughter of the Gladden Fields the Men of Westernesse were diminished, and their city of Annúminas beside Lake Evendim fell into ruin, and the heirs of Valandil removed and dwelt at Fornost on the high North Downs, and that too is desolate. Men call it Deadman’s Dike, and they fear to tread there. For the folk of Arnor dwindled, and their foes devoured them, and their lordship passed, leaving only green mounds in the grassy hills.’
The Fellowship of the Ring II.2.
Although Gondor flourished at first, eventually “the line of Meneldil son of Anárion failed” and the kings were replaced by ruling Stewards, of whom Denethor was the last. This means that Aragorn, who “is descended through many fathers from Isildur Elendil’s son”, is now the heir to the crown of Gondor and of Arnor, a claim that he hopes to use to establish feudal rule over the former territory of Arnor.
What is this feudal rule that Aragorn hopes to establish? Feudalism was a form of government in medieval Europe that operated by personal relationships of fealty. A feudal lord owned a territory that was too large to administer in person, and there was no system of civil administration on which he could depend, so he installed trusted vassals as tenants on parts of his territory. The vassals swore fealty to their liege-lord and there was a mutual relationship of support and security between them: the vassals were expected to support their lord by providing military service, and the lord was expected to support his vassal through gifts of land and treasure, and to protect the vassal if he was threatened by invasion, raids, or rebellion.
Describing the origin of the feudal system in Europe after the fall of the Roman Empire, Marc Bloch wrote:
[Charles Martel and his successors] wanted to establish order and Christian peace through their reams. They wanted soldiers to spread their dominion far and wide and to carry on the holy war against the infidel, an enterprise both conducive to the growth of their own power and beneficial for souls.
The older institutions appeared inadequate for this task. The monarchy had at its disposal only a small number of officials: but these were in any case not very reliable men and—apart from a few churchmen—they lacked professional tradition and culture. Moreover, economic conditions precluded the institution of a vast system of salaried officals. Communications were slow, inconvenient and uncertain. The principal difficulty, therefore, which faced the central government was to reach individual subjects, in order to exact services and impose the necessary sanctions. Thus there arose the idea of utilizing for the purposes of government the firmly established network of protective relationships. The lord, at every level of the hierarchy, would be answerable for his ‘man’ and would be responsible for holding him to his duty.
Marc Bloch (1939). Feudal Society, volume 1, p. 157. Translated by L. A. Manyon (1961). University of Chicago Press.
Thus vassals governed their own territories by subinfeudation, appointing their own sub-vassals to form a feudal hierarchy. This system was held together by personal relationships: ties of family and marriage, campaigning together in times of war, hunting, feasting, gift-giving, and so on. Vassals sent their sons to serve with their lord, and their daughters to serve their lord’s wife, in order to acquire these connections.
As kings they had to get supporters and above all create an army. So they attracted into their service—frequently in return for gifts of land—many men who were already of high rank. Former members of the military following, establishied on property granted by the ruler, did not cease to be regarded as his vassals; and his new followers were considered to be bound to him by the same tie, even if they had never been his companions-in-arms. Both groups served in his army, followed by their own vassals, if they had any. But, since most of their time was spent away from their master, the conditions under which they lived were very different from those of the household warriors of but a short time before. Each one of them was the centre of a more or less widely scattered group of dependants whom he was expected to keep in order; if necessary, he might even be required to exercise a similar supervision over his neighbours. Thus, among the populations of the vast empire, there became distinguishable a relatively very numerous class of ‘vassals of the Lord’—that is, ‘of the Lord King’ (vassi dominici). Enjoying special protection of the sovereign and being responsible for furnishing a large part of his troops, they also formed, through the provinces, the links of a great chain of loyalty.
Bloch, pp. 158–159.
Government of Arnor
With this model in mind, we can see what is happening in the exchange between Aragorn and Pippin. Aragorn has just been crowned king of Gondor, and he also claims the crown of Arnor by his descent from Isildur. In Gondor, there seems to be a well-established system of feudal loyalty, as we can see from the description of the feudal levy summoned by Denethor:
The men of Ringló Vale behind the son of their lord, Dervorin striding on foot: three hundreds. From the uplands of Morthond, the great Blackroot Vale, tall Duinhir with his sons, Duilin and Derufin, and five hundred bowmen. From the Anfalas, the Langstrand far away, a long line of men of many sorts, hunters and herdsmen and men of little villages, scantily equipped save for the household of Golasgil their lord. From Lamedon, a few grim hillmen without a captain. Fisher-folk of the Ethir, some hundred or more spared from the ships. Hirluin the Fair of the Green Hills from Pinnath Gelin with three hundreds of gallant green-clad men. And last and proudest, Imrahil, Prince of Dol Amroth, kinsman of the Lord, with gilded banners bearing his token of the Ship and the Silver Swan, and company of knights in full harness riding grey horses; and behind them seven hundreds of men at arms, tall as lords, grey-eyed, dark-haired, singing as they came.
J. R. R. Tolkien (1955). The Return of the King, book V, chapter 1. London: George Allen and Unwin.
Aragorn has secured his hold over these lords of Gondor through his bravery and leadership in the War of the Ring. But in the case of Arnor his rule is much less secure: the Rangers of the north are his loyal kinsmen, but Arnor is a vast land. Aragorn needs all the help he can get while he cements his rule by rebuilding the strongholds of Annúminas and Fornost Erain and installing loyal vassals there. A peaceful and loyal Shire will be an useful bulwark of the kingdom during this period of uncertainty, and so he reminds Pippin of the oath of fealty he swore:
‘Here do I swear fealty and service to Gondor, and to the Lord and Steward of the realm, to speak and to be silent, to do and to let be, to come and let go, in need or plenty, in peace and war, in living or dying, from this hour henceforth, until my lord release me, or death take me, or the world end. So say I, Peregrin son of Paladin of the Shire of the Halflings.’
The Return of the King V.1.
But as discussed above, the obligation goes both ways: if Pippin, as a vassal of the king of Gondor, is obliged to provide service to his lord, then Aragorn, as Pippin’s liege-lord, is obliged to support him in turn. It seems very likely that after the coronation of Aragorn, his vassals, including Pippin, swore fealty to him, and Aragorn swore his part of the obligation using similar words to those of Denethor:
‘And this do I hear, Denethor son of Ecthelion, Lord of Gondor, Steward of the High King, and I will not forget it, nor fail to reward that which is given: fealty with love, valour with honour, oath-breaking with vengeance.’
The Return of the King V.1.
This two-way relationship means that Aragorn, in reminding Pippin of his duty, also confers on him the honour and status as the personal representative of the king. We can see hints of this status in VI.6, where on the journey north “Pippin rode with the knights of Gondor”, establishing personal relations with his peers of the realm. We can see Pippin making use of this status when he returns to the Shire:
‘I am a messenger of the King,’ he said. ‘You are speaking to the King’s friend, and one of the most renowned in all the lands of the West. You are a ruffian and fool. Down on your knees and ask pardon, or I will set this troll’s bane in you.’
The Return of the King VI.8.
In the appendices we find that Aragorn uses his personal relationships with the members of the fellowship of the ring to help cement his control of the Shire and the lands to the west of it (the Westmarch):
1434 Peregrin becomes the Took and Thain. King Elessar makes the Thain, the Master† and the Mayor‡ Counsellors of the North-kingdom. […]
1436 King Elessar rides north, and dwells for a while by Lake Evendim. He comes to the Brandywine Bridge, and there greets his friends. He gives the Star of the Dúnedain to Master Samwise, and Elanor¶ is made a maid of honour to Queen Arwen.
1451 Elanor the Fair marries Fastred of Greenholm on the Far Downs.
1452 The Westmarch, from the Far Downs to the Tower Hills, is added to the Shire by the gift of the King. Many hobbits remove to it.
1462 At his [Sam’s] request the Thain makes Fastred Warden of Westmarch.
† Merry, the Master of Buckland. ‡ Sam, the Mayor of the Shire. ¶ Daughter of Sam and Rose.
The Return of the King, Appendix B.
Note Aragorn’s use of gifts of treasure (the Star) and land (the Westmarch) to his vassals, and his recruitment of Elanor (his vassal’s daughter) to his court to ensure that these relationships persist into the next generation. Elanor’s husband is then appointed lord of the Westmarch, extending Aragorn’s control over this region of Eriador through ties of loyalty and kinship.
The question wonders what motivated Pippin to offer his fealty to Denethor. But Pippin explains this clearly: it is a repayment of a debt he owes to Boromir, who died at Amon Hen in an attempt to save the hobbits. He cannot repay the debt to Boromir, but he can repay it to Boromir’s father.
Pippin flushed and forgot his fear. ‘The mightiest man may be slain by one arrow,’ he said; ‘and Boromir was pierced by many. When last I saw him he sank beside a tree and plucked a black-feathered shaft from his side. Then I swooned and was made captive. I saw him no more, and know no more. But I honour his memory, for he was very valiant. He died to save us, my kinsman Meriadoc and myself, waylaid in the woods by the soldiery of the Dark Lord; and though he fell and failed, my gratitude is none the less.’
Then Pippin looked the old man in the eye, for pride stirred strangely within him, still stung by the scorn and suspicion in that cold voice. ‘Little service, no doubt, will so great a lord of Men think to find in a hobbit, a halfling from the northern Shire; yet such as it is, I will offer
it, in payment of my debt.’ Twitching aside his grey cloak, Pippin drew forth his small sword and laid it at Denethor’s feet.
The Return of the King V.1.