The trick is using a conventional jet engine (as opposed to a (sc)ramjet), but cooled with the cryogenic rocket fuel. In effect, this is using the fuel as a heat sink, then expelling the fuel in the jet or rocket exhaust.
It's not a conventional jet engine - it has an intake turbine, but it's powered by the same helium cycle used by the precooler, and not by combustion byproducts.
The compressed air (or liquid oxygen when in rocket mode) and hydrogen is then combusted in a way similar to a conventional rocket engine. REL calls it an "air-breathing rocket engine", but it shares more in common with a ramjet than a conventional turbojet.
