"for the blood is the soul": This is the rationale for the mitzvah.
"and you shall not eat the soul with the flesh": This is eiver min hechai (a limb torn from a living animal). (Why is a verse needed for this?) Does it not follow a fortiori? Viz.: If (eating) meat with milk, which is permitted to a Noachide is forbidden to an Israelite, then eiver min hechai, which is forbidden to a Noachide, how much more so should it be forbidden to an Israelite! — (No,) this is refuted by yefath toar (a woman of beautiful form [viz. Ibid. 21:11-12]) and all like it, being forbidden to a Noachide, yet permitted to an Israelite. Do not wonder, then, if eiver min hechai, even though it is forbidden to a Noachide, would be permitted to an Israelite. It must, therefore, be written "and you shall not eat the soul with the flesh" — This is eiver min hechai.