the wedding ceremony, and Hippolyta is killed in the attack. Since Hippolyta is alive and well at the start ofA Midsummer Night's Dream, we can eliminate this version of the myth from consideration.

A more romantic version of the myth is that Hippolyta was in love with Theseus before he attacked the Amazons, and she betrayed the Amazons by running off with him. We'd like to believe this version of the myth—because this is a comedy, not a tragedy—but Theseus tells Hippolyta very near the beginning of the play:

THESEUS: Hippolyta, I woo'd thee with my sword,
And won thy love doing thee injuries (1.1.17–18)

which brings us back to one of the Theseus-defeats-the-Amazons-and-abducts-Hippolyta versions of the myth, although we're led to believe that, somewhere along the way, Hippolyta actually fell in love with Theseus. In other words, Shakespeare made up his own version of the story, which is something he did with just about every play he wrote. (The Tempest is Shakespeare's only entirely original play.)

At the start the play, Hippolyta echoes Theseus's feelings about their upcoming marriage:

THESEUS: Now, fair Hippolyta, our nuptial hour
Draws on apace; four happy days bring in
Another moon; but, O, methinks, how slow
This old moon wanes! She lingers my desires,
Like to a step-dame or a dowager,
Long withering out a young man's revenue.

HIPPOLYTA: Four days will quickly steep themselves in night;
Four nights will quickly dream away the time;
And then the moon, like to a silver bow
New-bent in heaven, shall behold the night
Of our solemnities. (1.1.1–11)

Hippolyta doesn't seem overly enthusiastic about the wedding, but she goes along with Theseus's happy anticipation of the "solemnities."

Nowhere in the play does Hippolyta actually say why she wants to marry Theseus. She says only that she's looking forward to marrying him, generally, and that she has a good feeling towards love, even after she hears about the merry mixed-up love of Hermia, Lysander, Helena, and Demetrius:

HIPPOLYTA: But all the story of the night told over,
And all their minds transfigured so together,
More witnesseth than fancy's images,
And grows to something of great constancy,
But howsoever strange and admirable. (5.1.24–28)

In trying to decide why Hippolyta wants to marry Theseus, we need to remember that weddings at the end of Shakespeare's comedies were intended to solve any and all problems that arose between a couple during the play. We can assume then that, no matter why Hippolyta wanted to marry Theseus, they lived happily ever after, which is what really matters.

In the opening scene of the play, Theseus is celebrating the fact that his "nuptial hour," the moment at which he will wed Hippolyta, is near. He reminds Hippolyta that he "woo'd thee with my sword," meaning that, essentially, he won her in battle—or, arguably, he impressed her with his skill. However, Theseus accepts that he has done Hippolyta "injuries" which are probably injuries to her pride rather than physical wounds. His suggestion that he will wed her with "pomp" and "triumph" indicates that he is attempting to change the "key" of their relationship and console Hippolyta, who may not be too happy about having been wooed in such a fashion. Hippolyta is very reticent throughout this first scene, not responding to Theseus's direct addresses, which confirms this.

Once an Amazon and "warrior," then, Hippolyta now faces the prospect of becoming Theseus's queen, but does not seem to face it happily.

Theseus has conquered Hippolyta's Amazon nation and has, in essence, "won" her.  Hippolyta, it is argued by some critics, is not extremely happy about her marriage to Theseus because she remains silent through much of Act I.  I can imagine she wouldn't be too happy about having to marry a man who conquered her people: 

They believe that this silence does not reflect Hippolyta's happy acceptance of her marriage to Theseus. Rather, her reticence suggests that she has been coerced into the marriage (remember, she has been taken captive), and that she seems to regard it with resignation and sadness.

The answer to your question on how the characters of Theseus, Duke of Athens, and Hippolyta, Queen of the Amazons, come to be engaged in William Shakespeare’s comedy A Midsummer Night’s Dream may be found in lines 16-20 of act 1, scene 1 of the play. Following his instructions to Philostrate, Master of Revels, to “awake the pert and nimble spirit of mirth” and make preparations for the wedding feast that is to take place in four days, Theseus says the following to Hippolyta:

Hippolyta, I wooed thee with my sword
And won thy love doing thee injuries,
But I wed thee in another key
With pomp, with triumph, and with reveling.

A bit of background information on Shakespeare’s source for the characters of Hippolyta and Theseus will help you to understand the meaning of these lines.

Hippolyta appears in Greek mythology as a daughter of the god of war, Ares. She is queen of the Amazons, a tribe of fierce warrior women believed to live in Asia Minor. Hippolyta wears a magic girdle (a kind of waist belt), given to her by her father, as a symbol of her sovereignty. She is connected in mythology to both Heracles (Hercules) and Theseus, the legendary king, hero, and founder of Athens.

In one particular version of the story of the ninth labor of Heracles, Theseus accompanies him on a mission to retrieve the girdle of Hippolyta. Hippolyta engages in battle but falls in love with Theseus and goes off to Athens to marry him.