Traditionally, the pilot controls are directly connected to all control surfaces, by hydraulics. FBW replaces this direct (and heavy) link with wires that merely transmit the information, not the mechanical movement, to actuators and motors at the relevant surfaces. The advantage is, the computer can be put in-the-loop. That means the computer can prevent the pilot from doing undesirable things (such as stalling, or pulling up in a stall). It also means, the computer can make the plane react to pilot inputs differently than it would if there was a direct link.
For example, if you use full deflection of some surface at high speed flight, the plane might react very violently, even tear off the surface in question, or produce structural damage. With a computer there, the deflection of the pilot control could be proportional to the rate of change of the flight path, as opposed to being proportional to the deflection. So moving the stick XX degrees to the left might produce the same result at any speed, whereas doing the same thing with a direct link would produce different results (less violent at low speeds, much more violent at high speeds). This is all theoretical - I do not know what exactly the computer does, it might well try and simulate the real thing very closely. Or it might not. One aim, apparently, is to make all planes feel the same to the pilot, so that an A320 is as sluggish as an A340 in its response, or vice versa, the 340 as lively as the 320. This reduces pilot type rating conversion times, along with similar cockpit layouts.
So, FBW allows the designer to let the plane response differently to pilot inputs than it would with direct links and hydraulics. It also allows the computer to have direct control and the final word (overruling pilot inputs) in controlling the plane.