Any lighting fixture is ultimately fed by the home's fixed wiring, which should be 12 or 14 gauge solid wire (usually in a cable form, with a black wire, white wire and a bare conductor).
Those 3 conductors are connected to your lighting fixture. In a flourescent fixture, the black and white wires will tie into the feed to a ballast, which will be smaller stranded wires that go to a black metal box with a label on it (the ballast). One wire from that ballast will go to each end of the lamp holders.
On your new incandescent fixture, those same 3 wires get connected. The fixture will have a white wire, a black wire and maybe a bare wire, or simply a green grounding screw. Connect fixture white to the white wire, fixture black to the black wire, and ground to the ground screw or ground wire.
It's quite simple and you shouldn't run into any surprises. There may be extra wires in the box, for switch loops or other items powered by the same circuit, but just pay attention to the wires that fed the ballast on your current cabinet, and use those. They may even be short "pig tails" tied into the more complex house wiring. In that case, re-use the pig tails.