What selective pressures favored an increase in brain size in hominid primates?

Basically, the increases of brain size is due to the environment. The brain developed by necessity, because primates where constantly facing new changes from the environment, unpredictable events, an constantly changing world obligates primates to evolve, otherwise they would die.

So, basically a complex world developed a complex brain, which implies a bigger brain. Until it became the modern, largest brain among primates, the human brain. This evolution gave the opportunity to process a lot more information, to deduct patterns in the nature, which was crucial to survival.

asthenosphere and convection