Berean Study Bible, Leviticus 17:
For the life of all flesh is its blood. Therefore I have told the Israelites, ‘You must not eat the blood of any living thing, because the life of all flesh is its blood; whoever eats it must be cut off.’
The operative word is H5315, nephesh, Strong's Concordance:
a soul, living being, life, self, person, desire, passion, appetite, emotion
The English word life does not cover all the nuances of the Hebrew word nephesh.
(14) For it is the life of all flesh . . . —Better, for the soul of all flesh is its blood, in, or through, its soul, that is, the sacredness of the blood arises from the fact that it contains the vital principle of all animal life. Or this clause may be rendered, for the life of all flesh is its blood in, or during, its life, that is, the life of all creatures consists in its blood; but only as long as the blood contains this life, for when it is dried up, or coagulated, the life has passed away from it.
Life is sacred; its blood is sacred.
What is the meaning of "the life of all flesh"?
It means the sacred soul-life of all animals.
How does it connect to the command that blood is not to be eaten?
You must not eat the blood of any living thing, because the soul-life of all flesh is its blood;
The sacred invisible soul-life is empowered by its physical blood. Eating its blood is like eating its soul-life.