The energy comes from virtual particles. You should think of it this way : anything has a certain chance of happening, but must follow conservation laws. Conservation of energy, conservation of mass, conservation of ... (the "real" conservation laws are more complex, but ... E.g. only mass+energy+entropy is constant, not mass, nor energy, nor entropy). But small, "short term" violations are possible.

Now imagine there are particle pairs that, when added together have zero mass, zero energy, zero ... These would pop into existence in pairs everywhere and be anihilated soon after. Same is true for triples, quadruplets, ... with different probabilities. Surprise : these particles "exist" (normally for very short times).

So particle a and particle b come into existence, both particles on a path that separates them from eachother. However, because opposite charges attract (and other effects related to other forces) they will fall back into eachother after a while, anihilating eachother and leaving nothing. These are generally referred to as "virtual" particles.

Except of course if something comes between them while flying on their path. This results in multiple effects, like the Casimir pressure, and hawking radiation.

The Casimir pressure is simpler. Suppose you have 2 straight plates, and you bring them close together. So close that these virtual particles have a good chance of impacting one but not the other. If the plates are metal, the particles will generally "merge" with it (like electrons). So a tiny percentage of the virtual particles between the plates become real.

This does not affect virtual particles that are not between the plates. So there is a clear differential between vacuum interactions outside of the plates and inside. This results in a massive pressure pushing the plates together, like removing gas particles in a container results in a pressure on the outside of the container (and, in an athmosphere of large molecules, if you move plates closer together than the size of the gas molecules, there will be a pressure that pushes the plates completely together).

Likewise, in black holes what comes between the particles is an event horizon. Event horizons "erase" quantum information, only the black hole as a whole has quantum information (the reason for is that black holes are physical, localized, manifestations of the end of time : they don't have an inside as far as anything outside the black hole can detect), no individual parts of a black hole have quantum state. So if an electron crosses the event horizon the charge of the entire black hole changes instantaneously, and the point charge of the electron disappears. That means the charge pulling the particles back together (assuming an e+ and e- virtual particle pair) becomes much more distant to the particle, and if it has enough speed and a convenient direction it will fly away from the black hole. Due to conservation laws the net result is that the black hole has eaten a somehow negative-mass charged particle, and the mass has decreased. A particle has been created outside of the black hole and is flying away to infinity.

The vast majority of these particles are photons of a specific frequency, and, surprisingly, the smaller the black hole, the larger the effect (and the higher the frequencies). So spaceships would have to carry "heavy" black holes to get a decent output and would have to "feed" the black holes to avoid exploding (millions of tonnes as least, and you have to shovel in matter at a rate of 10+ kg per second to keep it stable. Or you could reflect hawking radiation back in). And of course, large black holes have large inertia (plus : how do you hold on to one ? Electrical fields can do it, but it would have to be a field so huge it baffles the imagination ...).

A 1kg black hole would be converted into energy in less than a planck second by this effect and there is nothing, even theoretically, that can hold back that energy. Such an explosion might even do what people were "afraid" would happen with CERN's Higgs experiments and disrupt the Higgs field, causing a cascading conversion of the energy "powering" inertia into photons (the higgs field is something like a magnetic field that opposes all velocity changes in mass). If this were to happen, it would look one hell of a lot like a big bang.