How does a computer control electric circuits?
The computer produces a 'logic level' signal of a few volts at very low current. 

That can be amplified and used to operate a mechanical relay or a solid state relay to as high power as is required.

For example, the milliwatt range logic level output could drive a small transistor, and the transistor could drive a ~1W coil controlling a 30A relay for an appliance. The power to operate the relay would generally be supplied externally. The relay could switch 240VAC at 30A, which is over 7kW.