What they get their energy from? Well, fusion (and fission, in case of LACs) reactors, obviously! Duh!
But... there's a problem. Or rather - a missing step. You have a fusion/fission reactor. It fuses/fissures atoms, creating heat and radiation. Both are forms of energy, but in general, neither heat or radiation are useful to do anything with it (beyond heating/irradiating stuff). How do you convert that into electric (or whatever) energy drive nodes need? Or for that matter, grasers?
In our world, a fusion/fission reactor does this by heating water. The water is then fed through a turbine which powers a generator which generates electricity. So we convert thermal energy to kinetic energy and then to electric energy.
But how is this energy conversion done in a starship? For some reason I find it... hard to believe that a Manty superdreadnought or a Grayson LAC have steam turbines churning deep inside their engine rooms... not only does this appear to be a bit too anachronistic, I also seriously doubt that any turbine-powered generator would be able to generate enough electricity to power a Graser battery which is able to "shatter a small moon" (quote from the books).

Obviously there's some sort of magical generator device which is able to turn heat and radiation directly into electrical (or whatever) energy. Does anyone have a clue what that might be?