To add on to other peoples answers regarding the complete nutritional makeup of pet food; many animals can make a variety of the amino acids they need to survive with just a few inputs (like deer and cows eating only (mostly) plants), but some, especially predatory animals, cannot. They get those nutrients from the prey they eat, which in turn got them from the plants.

It essentially comes down to which enzymes any given organism can create, which ones their DNA codes for. Humans can't make a bunch of these amino acids themselves. Many (maybe all of them, not that far into my class yet) of the reactions taking in place in any living organism are entirely reliant on enzymes to catalyze them; that is, without them these reactions would take millions of years to complete.

BTW there are appr. 37x10^17^ (3,700,000,000,000,000,000) reactions happening in your body every second. All of them (or at least a great majority of them) require enzymes to complete.