Though its job is mainly to capture deuterium, I presume for propulsion. (In a fusion process? This is where Star Trek veers off into literary freedoms with the dilithium crystals and what not.)
IIRC, it’s a matter-antimatter drive, and the dilithium crystals in some way help to catalyze the production of antimatter. As you say, fictional.
No, the dilithium is somehow supposed to help control the matter/antimatter reaction. The antimatter is stored onboard in magnetic bottles I think.