Hi Sam! Thanks for asking!
"Graspers" are Matthias and Geoffrey's name for the inhabitants of a particular simulated reality that Matthias allowed to "graduate" when they weren't "ready" for it. They were heirarchical and dominance-driven and warlike (like, uh, us), and eventually Matthias had to violate his principles and imprison them back in their simulation, a fact he tried not to think about much. This is of course Matthias and Geoffrey's version of things -- the "Graspers" might differ. I'm a little uncomfortable, myself, with the whiggish notion that societies intellectually capable of figuring out that they are in simulation are also "mature" or "advanced" enough to have done away with brutish and dangerous forms of competition -- I would rather suspect the opposite, some days -- but Matthias seems to have gotten away with this ideology so far.
Basically, think of Graspers as some cooped-up violent parakeets stuck in Matthias's library.
"Ontotropes" are made-up physics having something to do with a multiverse model, sort of similar to string theory, in which there are other universes than ours displaced along other dimensions than the ones we know.
From the original prologue:
The world we see is woven out of strings, and they of branes, and they of monads, and monads out of transpelagic omniplexic theremons, and theremons of ontotropes. And there among the ontotropes, transverse to the space we know, where time runs sideways and causality tiptoes around knots of killing paradox, there is another house.