אך בשר בנפשו דמו לא תאכלו-But flesh with its soul its blood you should not eat
Rashi explains this Pasuk as referring to the prohibition of אבר מן החי – eating A limb taken from a living animal. He explains that even though Noach had been given permission to eat meat, this was putting a limit on this privilege and not allowing people to tear limbs from living animals. The Emes Liyakov (Harav Yaakov Kamenetzki) explains why this comes specifically after the flood. He explains that if we look at the previous Pasukim, Hashem “smells” the scent of Noach’s offering, and because of this decides never to bring a flood again because of the sins of man. With this in mind, Hashem needed a way to insure that the situation in the world would never again become as evil and corrupt as it had become during the period before the flood. To that end, He commanded Noach not to eat a limb from a living animal. Specifically, this command served this purpose because all the other laws for the descendants of Noach would probably be established even if Hashem hadn’t commanded them as they are universally logical (e.g. prohibition against killing or stealing). Through keeping the other commands of Hashem given to Noach’s descendents, people wouldn’t necessarily recognize them as commandments of Hashem, as they might have thought that they were following rules developed from their own logic. Therefore, Hashem commanded a mitzva that a person might not logically enact. By doing this, every time a person would refrain from eating a limb from a live animal, he would remember Hashem’s power in maintaining the world.
With this explanation, the Emes Liyaakov explains the continuation of the Parsha in which Hashem tells Noach that the rainbow will be the sign that He will not bring another flood. Here, as well, Hashem was trying to insure that the world would never sink to the level of evil that would necessitate bringing another flood to change the entire nature of the world. If we look at Gemara Sanhedrin, it tells us that the generation of the flood was extremely successful and prosperous. This is because Hashem was not concerned with punishment – if He would need to destroy the world, because man did not keep the world properly, He would destroy it. After the covenant He made with Noach, just destroying the world and starting again was no longer an option. Hashem needed to establish cues so that people would realize that when Hashem brings good to the world even though people don’t deserve it, they need to recognize this kindness in order to repent and not bring the world down to the level of deserving another flood. To do this, Hashem put a sign in nature that would serve as this reminder. We, as well, need to remember that Hashem is constantly trying to bring us closer through mitzvos and actions that will keep the world at a higher level.