Here's Andrew Probert's descripction of the impulse deflection crystal:
"In order to power the impulse engine while the Enterprise
was operating as a single unit, the thinking was that we would channel energy from the main reactor, if you will, up into the crystal, which would deflect that energy into the impulse engine units. So it was literally that -- a crystal that would deflect power into the impulse system. I didn't have much of a handle on how the ship would be powered at that time, which didn't really come into fruition until Rick Sternbach helped me sort that all out on Next Gen
What Andrew Probert described, sounds a lot like an antimatter rocket
where the heat from the matter-antimatter annihilation is channeled to the crystal which then heats up the impulse fuel and results in an impressive particle thrust.
Alternately you could get similar results from a fusion rocket, where - in essence - the engine nozzle is like a valve of a fusion reactor, releasing the hot ion plasma to achieve particle thrust.
Apparently, the dilithium crystals transform the matter-annihilation products into some kind of exotic energy or radiation that can be used to power the warp engines. It seems there's still not that much more speculation as to how the dilithium crystals could actually work.
Of course there's one blatant error I'd like to see fixed ASAP: Spock recrystalizing dilithium by exposing it to gamma radiation.
Gamma radiation is the biggest product of matter-antimatter annihilation...