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.
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.
In the test rig, it's being cooled by boil-off of LN2. In a flight configuration, using cryogenic propellants as a heat dump seems logical.