We don't know, but it (apparently) does not use fuel. According to the in-game codex (secondary entry "Reaper Capabilities"):
The Reapers' thrusters and FTL drives appear to propel them at more than twice the speed of Citadel ships. Estimates of their location in dark space suggest they can travel nearly 30 light-years in a 24-hour period.
Reaper power sources seem to violate known physical laws. Reapers usually destroy fuel infrastructure rather than attempting to capture it intact, indicating that Reapers do not require organic species' energy supplies. Consequently, the Reapers attack without regard for maintaining supply lines behind them, except to move husks from one planet to another. Unlike Citadel ships, Reapers do not appear to discharge static buildup from their drive cores, although they sometimes appear wreathed in static discharge when they land on planets.
Presumably, the Reapers acquire and build up energy by some unknown means, most likely in dark space between harvests. I would then tend to postulate that, if they are not releasing any reaction mass, they must be using reactionless drives, like the Tantalus Drive Core in the SSV Normandy (but much more advanced), which is described in Mass Effect's version of the codex as follows (in the primary entry for the ship as a whole):
Another component of the stealth system is the Normandy’s revolutionary Tantalus drive, a mass effect core twice the standard size. The Tantalus drive generates mass concentrations that the Normandy 'falls into', allowing it to move without the use of heat-emitting thrusters.