Based on game mechanics (I know, I know, mechanics =/= lore), I would imagine that due to silence preventing casters from casting spells, you can infer that there is a verbal component to casting a spell while/after casting it (i.e a black mage channels the aether/mana it needs to cast flare then says "FLARE!" when it finishes casting). additionally the caster's weapons may act like a focus for their spells (I believe the relic and anima weapon quest touch on this but it's been so long since I've done them that I don't remember the details). Also the soul stones probably offer the user some knowledge on how to use their more advanced spells and abilities.

I'm sure some of the other more knowledgeable lore masters could provide better insight to this, and the lore book probably also goes into detail about how the other jobs beside arcanists, scholars, and summoners cast their spells.