The energy is given off in the form of electromagnetic waves. The energy ultimately comes from whatever process is accelerating the charge. That is, because some of the energy gets radiated away, you need to do more work to accelerate a charged object than an equivalent uncharged object. Thus we speak of a "radiation reaction force" that opposes the acceleration of charged particles. We can say that the energy of the radiated EM waves comes from the work you do against this radiation reaction force.
(Since you posted this in the QM forum, you might note that this is a purely classical phenomenon, not a quantum mechanical one.)