Why exactly does Kyoko attempt to save Sayaka?
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.