It doesn't say Man, it says "Flesh". Both man and animals have flesh, therefore it applies to both. It is very specific in that it says all flesh, and not "the flesh of man". It says "Flesh with the life thereof", both animals and humans have flesh, both animals and humans have life. Simple as that. Both animals and humans have spirit/soul. "Spirit" is just another word for Consciousness, consciousness is an effect of the soul. Though there is no known modern "scientific" proof for the soul, the way to verify the presence of the soul is if there is consciousness (spirit).
Both animals and humans have flesh, and spirit/soul, and therefore life. This verse clearly is an instruction to be vegetarian.
(I may or may not comment here anymore.. I don't know.. that just had to be cleared up.)