star trek - What does it actually mean to travel in trans-warp?

In the TNG Episode "Descent, Part I" Geordi LaForge makes it clear that the term "Transwarp" specifically refers to any technology that would allow a ship to exceed the "normal subspace limitations" that prevent Starfleet vessels from travelling beyond Warp 9.999

GEORDI : Our current theory is that the Borg have established several transwarp conduits through subspace. A ship entering a conduit
is immediately accelerated to an extremely high warp velocity. It's
like falling into a fast-moving river and getting swept away by the
current.

PICARD : How fast would a ship travel through one of these conduits?

GEORDI : We don't know... normal subspace limitations don't apply to transwarp variables. Based on the distance we covered in our trip
through the conduit, I'd say the speed is at least twenty times faster
than our maximum warp.

In the Voyager episode "Threshold", Tom Paris manages to exceed the Warp 10 barrier and achieves 'transwarp speeds' (albeit with some additional mumbo-jumbo that's never spoken about again).

KIM: Nothing in the universe can go warp ten. It's a theoretical impossibility. In principle, if you were ever to reach warp ten, you'd
be travelling at infinite velocity.

NEELIX: Infinite velocity. Got it. So that means very fast.

PARIS: It means that you would occupy every point in the universe simultaneously. In theory, you could go any place in the wink of an
eye. Time and distance would have no meaning.

In the final episode of TNG an alternative future version of the Enterprise-D is able to reach transwarp speeds in excess of "Warp 13"

In answer to your specific question, 'how can Voyager travel in Transwarp?', the answer is that they've stolen a "transwarp coil" from the Borg. The actual technical details are never described.