In Astronomy Cast Ep: 74 they mentioned that anti-matter doesn't exist within "anti-time." And in fact, the female voice of the show says that anti-time doesn't exist (she jokingly says that anti-time is running the clocks backwards). The problem I have is grasping the non-existence of anti-time when Richard Feynman through his diagrams shows that antimatter particles exist backwards in time.

I don't entirely understand what that means, but the way I interpreted it is that the existence of the particles is already laid out. Think of a time line (flowing left to right as time progresses). At point 1 is the creation of matter and anti-matter. (0-1 exist only energy in the form of photons) At point 10, the two particles collide and points 10-11 is the same photon from segment 0-1. So it looks like this ----==========----. The equal sign is the matter and antimatter particles, the dash is the photon. So a normal particle exist left to right. An antimatter particle exist right to left.