St. Thomas Aquinas was a 13th century philosopher, theologian, and priest. He is probably best known today for his five proofs of the existence of God, although he most likely would have considered them only a small part of his philosophy (and many of them predated him). The comic is mostly parodying the Ontological Argument, (which Aquinas actually rejected on the basis that we cannot know the nature of God. We can, however, know the nature of hats, so there is no reason to believe he would object to it here).
Malebranche was a 17th century philosopher and theologian, whose main project was to synthesize the theology of St. Augustine and the philosophy of Descartes. A big problem in Cartesian philosophy was the problem of interaction, that is to say: how does the mind interact with the body? Since the material world appears to be causally complete, it seems impossible for the mind to affect anything. Malebrache's solution was "Occasionalism", which said that there was no interaction per se, but God would go in each and every time and sync up the material and spiritual world. So if I attempt to raise my arm, it would only raise if God raised it, since my mind can't affect the world.
Jean Buridan was a 14th century philosopher and priest, best known for the thought experiment making fun of his mental determinism, "Buridan's Ass", in which a donkey can't decide between two bales of hay because it perceives them as exactly equal (he thought our minds would always choose the best course of action).
Saint Augustine was a highly influential philosopher and theologian in the fifth century, although none of his specific ideas really come up in the comic.