How does one internalize many theorems quickly?

Hm, interesting. Only did a few math competitions myself, which I mildly regret; didn't have anyone to push me in that direction growing up--ah well, that's neither here nor there. I imagine it would be quite difficult to internalize those theorems from just the IMO Compendium's nice summary. They would mostly be disconnected facts with very little, probably too little, context.

To pick some random examples, Theorem 2.13 on solving linear recurrence relations by factoring a polynomial is more summarizing the result of the standard approach to such problems (off the top of my head, it would certainly be discussed in Wilf's very readable generatingfunctionology, perhaps somewhat less generally) than giving you the approach. Similarly, it gives both Langrange's theorem and Fermat's Little Theorem without mentioning that the latter is a corollary of the former or the importance of the group structure of Z/pZ to the whole thing. Theorem 2.127 gives the standard parameterization of Pythagorean triples, but it's not connected to Theorem 2.98 which gives the rational parameterization of the unit circle and which is the key tool for actually proving the standard parameterization. It's the idea of the parameterization of Pythagorean triples rather than the result that you'd probably be expected to generalize in a contest problem anyway.

So, I suppose I'd slog through the problems themselves, reading plenty of solutions and trying a good chunk yourself without help (maybe with a self-imposed time limit, so you don't get stuck). If you find you're often running up against the use of theorems from the list, then you'd have a clear reason to internalize a relevant subset. You might also want to look up more information (e.g. reading the Wikipedia page on Fermat's Little Theorem) to supplement descriptions of results which are too sketchy to grab hold of.

Best of luck!