Everyone has seen a military drill instructor (DI) marching next to a platoon of recruits. By using verbal commands, he can direct the speed, direction, and even the "style" of the marching soldiers. He can do this because he can see the soldiers, and can choose the moment that he will call for a change in their direction of travel. He can see how effective his commands have been, and can make corrections as necessary, by issuing new commands.
The controller can see you on his radar screen. He knows where the VOR's, intersections, airports, etc are located, and can direct you by naming a magnetic heading for you to take up. Of course, he can also see the enhanced returns (radar echos) of the other aircaft on his screen, too. If he is providing separation to the IFR aircraft, he may ask you, if in radio contact, to make a turn in order to provide that IFR separation to the IFR aircraft.
It's as if the controller is "steering" your aircraft by remote control.