If you understand that GTX is a super thin membrane with about the thickness of cellophane, you get a better sense of how it works and impacts breathability. As others have said, it's just a membrane filled with millions of tiny holes. In a boot, it's really like waterproof underwear, since it's used in the inner part of the boot. I think some people sometimes confuse breathability and membranes in general. If you wear two identical boots, one with, and one without a GTX liner, you might find the GTX boot slightly less breathable, but the difference really should be slight. What I find is people will say their light, mesh clad non-GTX boot breaths better than their 5mm full leather boot with GTX. Well, duh. Not that the GTX is the big factor.
One of the really big factors with GTX boots is relative to socks. Cotton socks in GTX boots is a disaster. GTX requires foot moisture to be vapor in order to pass through it. If you have a sock that does not permit appropriate wicking action, the moisture can't be transformed to vapor to pass through the membrane. Some socks pool moisture and that moisture will not pass through a GTX, eVent or other membrane very well.
By the way, even the footbed can be a big factor since the majority of your foot's moisture exits via the bottom of your foot. If the footbed is overly hydrostatic, it too will pool moisture and give you nice hot-box feet.