How do I explain my mother's leg amputation to my daughter?

Saved her life. Hopefully your daughter is old enough to understand death. And it would seem obvious that a treatment as severe as an amputation was necessary to save your mother's life. "the doctors saved grandma's life." will show her how important it is. The second part, getting her to be comfortable with it, is all on you...You have to set an example and be ok with the new deformity.
Age. Well it depends on the child's age. Usually best is to explain with the truth. Grandma had a leg that was sick and was not working well, so it had to be removed.