The principle behind antimatter is simple. It releases a ton of energy when it's destroyed, so you'd harness that in the same way you'd harness energy from nuclear fusion or any other "atoms smashing it up" style energy. It sounds like science fiction, but in fact most of the gorgeous renderings of antimatter spaceships you'll find out in the world are actually done by NASA artists or people in labs.

Here's a cool diagram of the ICAN, also dreamt up by the Penn State antimatter propulsion lab:

A strangely thrusty antimatter ship from NASA:

And an antimatter ship designed by NASA to go to Mars: Hey ho, let's go!