In A Midsummer Night's Dream, Theseus tells Hippolyta:
THESEUS:
Hippolyta, I woo'd thee with my sword
And won thy love doing thee injuries,
But I will wed thee in another key,
With pomp, with triumph, and with reveling.
Why does he say this? How is it possible to woo someone with a sword? Why did Hippolyta give him her love in return for injuries?
I have unfortunately run into obstacles right at the outset.
Lauren Ipsum's answer is correct, as far as the meaning in the play itself goes, but it's also worth noting that Shakespeare's portrayal of Theseus and Hippolyta was based on a much older story.
There are many variations of this story in different retellings of Greek mythology. Sometimes Theseus marries Hippolyta, sometimes another Amazon woman; sometimes they go to war again afterwards and she ends up being killed. But the stories seem to agree that there was a war between Theseus (possibly together with Heracles) and the Amazons (led by Hippolyta). From Wikipedia:
In the myth of Theseus, the hero joined Heracles in his expedition, or went on a separate expedition later, and was actually the one who had the encounter with Hippolyta. Some versions say he abducted her, some that Heracles did the abducting but gave her to Theseus as spoils, and others say that she fell in love with Theseus and betrayed the Amazons by willingly leaving with him. In any case, she was taken to Athens where she was wed to Theseus, being the only Amazon to ever marry. In some renditions the other Amazons became enraged at the marriage and attacked Athens. This was the Attic War, in which they were defeated by Athenian forces under Theseus or Heracles. In other renditions Theseus later put Hippolyta aside to marry Phaedra. So Hippolyta rallied her Amazons to attack the wedding ceremony. When the defenders closed the doors on the attackers, either Hippolyta was killed, Theseus directly killed her in the fight, she was accidentally killed by another Amazon, Molpadia, while fighting by Theseus’ side, or was accidentally killed by her sister Penthesilea during this battle or in a separate incident. This killer was in turn slain by Theseus or Achilles. Some stories paint Theseus in a more favorable light, saying that Hippolyta was dead before he and Phaedra were wed, and this battle did not occur. Further complicating the narratives, a number of ancient writers say the Amazon in question was not Hippolyta at all, but her sister Antiope, Melanippe, or Glauce. Moreover, there are combined versions of the tale in which Heracles abducts and kills Hippolyta while Theseus, assisted by Sthenelus and Telamon, abducts and marries Antiope. There are also stories that Hippolyta or Antiope later bore Theseus a son, Hippolytus.
One particular recounting of the story, which matches the version of Shakespeare in which Theseus and Hippolyta first go to war and then get married, is that of Plutarch, "The Life of Theseus", part of his Parallel Lives:
Well, then, such were the grounds for the war of the Amazons, which seems to have been no trivial nor womanish enterprise for Theseus. [...] Now for a long time there was hesitation and delay on both sides in making the attack, but finally p63 Theseus, after sacrificing to Fear, in obedience to an oracle, joined battle with the women. [...] And after three months, he says, a treaty of peace was made through the agency of Hippolyta; for Hippolyta is the name which Cleidemus gives to the Amazon whom Theseus married, not Antiope.
Here Theseus goes to war with the Amazons and later marries Hippolyta, just as is suggested in A Midsummer Night's Dream. Plutarch's version of the story is based on that of Cleidemus, but the latter unfortunately appears to have been lost. However, the influence of Plutarch's writing on Shakespeare is well documented, and not restricted to A Midsummer Night's Dream. It seems to be generally accepted that Shakespeare's Theseus and Hippolyta are based on Plutarch's, who went to war and then were married.
Hippolyta is queen of the Amazons. (You may be more familiar with her as Wonder Woman's mother; she's the third woman from the left, next to Diana, in this photo.)
Amazons are known mythologically for being fierce warriors. To woo (court, date, interest) a warrior, you might spar, or fight. Theseus is saying that instead of courting her with flowers and poetry, he did it by battling her in a swordfight. He earned her affections by getting through her literal defenses and injuring her with his sword, proving he was an equal or better fighter. The queen of a warrior race is a mighty warrior herself, and isn't going to settle for someone who can't fight as a consort.
So Theseus says that while he won her with violence (fighting, injuries, which a warrior accepts as normal and possibly desirable to some extent), their wedding will be full of celebrating and fun.
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
