Protein and caloric density = people living longer and babies being more likely to survive to maturity.
Hunting = more planning than gathering, so as roger said, it favors intelligence. Particularly if early humans had a sense of fairness, quite literally, the hunter would get the spoils (e. g. more of the protein, etc.) and be more likely to survive longer.
Hunting also = it's more dangerous to get those calories and protein. Hence it also favors teamwork and stealth. I don't want to sound Lamarckian and that's not my intention. Rather, that, as said above, the hunter gets the spoils and has more calories and lives longer and can reproduce more.
More reproducing = more chances for a descendant to survive.
Because all evolution requires from your genes is two things:
- Live long enough to reproduce and
- Do so.
How you get there is immaterial. You just need to do both and your genes will stay in the pool.