Perhaps her motivation will become clear as you hear about her. Like all legends, the story of Morgan Le Fay changed over time (think Robin Hood and even King Arthur himself). The most commonly known information about her connects her to King Arthur, though earlier in historical and literary documents this was not so. Here's the gist of her story:
Morgan (or Morgana) is a witch who studied with the great Merlin. She is the half-sister of Arthur (they have a shared mother). She has two sisters, one of whom is Gawain's mother. She becomes a life-long adversary to her half-brother and his wife Guinevere. She consistently tries to impugn the honor of King Arthur and his knights (her motive in sending Bertilak?). She may or may not have thrown Arthur's famous sword Excalibur into the lake. She may or may not have had a child with Arthur, known as the evil Mordred.
This is the best that can be done with legends--take what you will from whatever time period you wish. In general terms, though, her name explains her character--fay means fairy, which is just a whisper away from witch.
According to the Arthurian legends, Morgan Le Fay is Arthur's half-sister. She is a priestess, and has spent much of her time at Avalon with the other mystical women of the time period. When Arthur came of age, he was expected to celebrate this rite of passage by mating with a priestess from Avalon. Unbeknownst to Arthur and Morgan Le Fay, the pairing for this rite, they are related. As a result of this pairing, the two produce Mordred, Arthur's illegitimate son, who is responsible eventually, for Arthur's death. The two are at odds with one another as Mordred is jealous and upset that he is not recognized as the King's son and treated with more pomp and circumstance.
Morgan Le Fay, a dark beauty, is angry with Arthur, too, because he prefers the fair beauty of Gwenivere to herself. So, in a way, both she and Mordred have been snubbed by Arthur. This is probably why she would have tested Arthur and his knights of the Round Table in terms of their honor, bravery, generosity, and willingness to protect those who are weak and unable to protect themselves (women, children, elderly, etc.).