Antimatter is only used because, when combined with matter, it has a higher energy density than anything else known, including fusion. That part is completely within known physics. The "dilithium" is used somehow to control the matter-antimatter reaction; they have nothing to do with the warp field. Why the writers came up with this I don't know, but it does make for a convenient plot device sometimes when they have some problem with dilithium mines on some distant planet. Like various hard-to-acquire materials today, there are political problems caused by the scarcity of dilithium, and this comes up in many episodes. And, conveniently, dilithium somehow can't be replicated by the regular matter replicators.

The warp field is created by the warp nacelles. How? Who knows.