In episode 8 of Puella Magi Madoka Magica, we see Sayaka transform into a witch, and in that episode and the subsequent one (at least in the second half of it) we see Kyouko's attempt to save her.
Why exactly does she seem so determined to do this, given that she'd talked earlier about living for oneself? (e.g. Perhaps out of sympathy for Sayaka's situation or a change of heart? Or maybe because of her "goals", which don't seem to be very well-specified when referred to in episode 9?)
Episode 7 paints the picture of Kyouko's background. Basically, Kyouko is all alone having lost everyone important to her including her family. So she's in despair and desperate to find comfort somewhere.
After hearing Sayaka's story, Kyouko started to identify with her. So they effectively formed a friendship. Episode 10 reveals that this friendship is formed in multiple timelines.
So when Sayaka becomes a witch, Kyouko felt the obligation to save her - her one and only friend.
