The context of the commandment to Noah is important. If you read it in context, it is not for the first time allowing meat as food, it is simply forbidding eating meat that is not slaughtered, meat which is still alive, with blood coursing through it's veins. I suppose this is why swallowing live goldfish was a popular fraternity hazing practice in the 1950s.
Anyway, here is the passage:
Bear fruit and multiply, and fill the land. And your peculiar feats will awe all the land animals and all the fowl of the sky. All that will crawl the land, and all the fish of the sea; into your hands they are given. Every land crawler that lives, will for you be for eating, of green plants, I give you all. But flesh, with its soul still in its blood you will not eat. And even the life-blood of your soul I will demand, by the hand of every animal, I will demand it, and by the hand of man, in the hand of a man and his brother, I will demand the soul of man. Who spills man's blood, will that man's blood be spilled: because in the image of God, is man made. And you, bear fruit and multiply; infest the land and be many there.
First, the permission is granted to eat plants, which is not a new thing. The "Bear fruit and multiply" is not new either. The new thing here is the commandment for killing killers and to not eat things that are alive.
The nature of this commandment is simple: don't murder and eat whatever you want, so long as you kill it first. It is this governance for non-Jews, to be contrasted with the more detailed Mosaic law that follows, which applies to Jews. The Mosaic law is very detailed regarding what one should do in this circumstance or that, the Noahide law is the religious version of anarchy.