They're pre-war tech so no post-war engineer will have a hundred percent acurrate answer. They're still produced/maintaned by Enclave Engineers though, who might have a pretty decent guess abou that.
My guess, which is as good as yours, is that they probably have an eletromagnetic field in the same polarity as the ground, with a possibility of switching poles in different grounds, expelling him from the surface, making him float. As you don't see any jets, steam or rocket propulsors coming aaout of it.
A series of ducted propellors, or small nuclear jets. Most jets don't have visible flame coming out of them, airliners, for example never do, unless they're crashing. EDE doesn't actually float, as you'll notice he can't pass terrain that's impassible to you.