As far as I can tell, there isn't even a treknobabble explanation for how the shroud works. It's evidently a biological function (since in DS9: The Abandoned, we see a young Jem'Hadar warrior who learns to shroud without any external support or the implantation of any technology) but one that results in the user's clothes and and held items also becoming invisible to the naked eye. This heavily implies that they're emitting some kind of field or particles like the Suliban cloaking particles seen in Ent: The Communicator. Beyond that, we simply don't know.
In Worlds of Star Trek: Deep Space Nine, Volume 3 it's explicitly stated that the ability to shroud is a facet of psychic ability, one that also (theoretically) allows the user to remotely view events happening outside of their eyeline.
As Taran’atar stopped at Matheson’s order, he felt his mental grip on
the image of the Founder begin to slip. Dimly aware of the door up
ahead blocking the way, and of the lieutenant commander pushing a key
into a slot in the bulkhead, he struggled to preserve his clear
visualization of the Founder. He did not know how much longer he could
sustain his efforts. Shrouding normally required a significant
exertion of will, but far less, it turned out, than the task at hand.
Taran’atar had never attempted what he did now, had not even known it
possible until a few moments ago, having heard only unconfirmed rumors
of Jem’Hadar who had utilized their shrouding capabilities for remote
generation of images.
Activity up ahead penetrated his awareness. He plainly heard Matheson
identify herself and request access to the defensive emplacement,
distinctly felt vibrations through the decking as the door began to
slide slowly into the bulkhead. Too far, he thought, intuitively
understanding that his capacity to project a realistic image relied
not only on his ability to concentrate, but also on his distance from
the location to which he projected that image.