Shakespeare took the characters of Theseus and Hippolyta from his sources, most likely Plutarch’s biography of Theseus and Chaucer’s Knight’s Tale.
But Shakespeare uses these characters in his own story, which does not correspond closely to Greek mythology. In the myth, Theseus, king of Athens, fought the Amazons and abducted an Amazon woman (Hippolyta in some accounts; Antiope in others), and this led to the Attic War. But in A Midsummer Night’s Dream, Theseus and Hippolyta are witty aristocrats presiding over a festival of love and marriage. We can’t expect to find psychological realism in this kind of mashup.
So when Shakespeare has Theseus say:
Hippolyta, I woo’d thee with my sword
And won thy love doing thee injuries,
he is simply alluding to the myth, which his audience could be assumed to be familiar with. Theseus fought the Amazons (“woo’d thee with my sword”) and abducted Hippolyta (“won thy love doing thee injuries”). By injuries here Shakespeare means “wrongful actions or treatment” and not necessarily “bodily wounds”.
There are psychologically realistic ways to interpret this: perhaps Hippolyta really did fall in love with Theseus and so her abduction was consensual. Or perhaps she is making the best of a horrible situation by allowing Theseus to believe that she loves him. But we can’t expect the text of the play to help us out.
In the biography of Theseus from Lives of the Noble Greeks and Romans, Plutarch has some trouble organizing the many different versions of the story of Theseus and the Amazons. In Thomas North’s translation (first edition 1579):
Philochorus, and some other hold opinion, that he went thither with Hercules against the Amazons: and that to honour his valiantness, Hercules gave him Antiopa the Amazon. But the more part of the other historiographers, namely, Hellanicus, Pherecides, and Herodotus, do write, that Theseus went thither alone, after Hercules’ voyage, and that he took this Amazon prisoner; which is likeliest to be true. For we do not find that any other who went this journey with him, had taken any Amazon prisoner besides himself. Bion also the historiographer, this notwithstanding, saith, that he brought her away by deceit and stealth. For the Amazons (saith he) naturally loving men, did not fly at all when they saw them land in their country, but sent them presents, and that Theseus enticed her to come into his ship, who brought him a present: and so soon as she was aboard, he hoised his sail, and so carried her away.
Moreover Plutarch’s sources disagree on the name of the Amazon woman that Theseus abducted:
For this historiographer [Clidemus] calleth the Amazon which Theseus married, Hippolyta, and not Antiopa.
Nonetheless, the outlines of the story are fairly clear: Theseus went to the land of the Amazons, fought there against the Amazons, and returned to Athens with a prisoner, who was either Hippolyta, queen of the Amazons, or her sister Antiope.
Shakespeare was familiar with North’s translation of Plutarch, and used it as a major source for the plays Julius Caesar, Coriolanus and Antony and Cleopatra. In A Midsummer Night’s Dream there are only passing indications that Shakespeare had consulted Plutarch, but here’s one in Act II Scene I:
OBERON. How canst thou thus, for shame, Titania,
Glance at my credit with Hippolyta,
Knowing I know thy love to Theseus?
Didst not thou lead him through the glimmering night
From Perigouna, whom he ravished?
And make him with fair Aegles break his faith,
With Ariadne and Antiopa?
North’s translation of Plutarch has the same spellings of these names of Theseus’ lovers: Perigouna for Perigune, Ægles for Aegle, and Antiopa for Antiope.
Shakespeare’s choice of ‘Hippolyta’, rather than ‘Antiopa’ as in North's translation of Plutarch, is likely to have been influenced by Chaucer. In the prologue to the Knight’s Tale, Chaucer gives a brief account of the myth, using the name ‘Ypolita’:
Whilom, as olde stories tellen us,
Ther was a duc that highte Theseus;
Of Atthenes he was lord and governour,
And in his tyme swich a conquerour,
That gretter was ther noon under the sonne.
Ful many a riche contree hadde he wonne,
What with his wysdom and his chivalrie;
He conquered al the regne of Femenye,
That whilom was ycleped Scithia,
And weddede the queene Ypolita,
And broghte hir hoom with hym in his contree,
With muchel glorie and greet solempnytee,
And eek hir yonge suster Emelye.
And thus with victorie and with melodye