I'm going to guess there will be no good introductory texts at that level that will be all that useful. You will probably have to delve into the physics literature (particularly, review articles). Although, a lot of lattice QCD stuff is written for experimentalists who know little formal QCD or math, so you should be able to find something reasonably accessible for a beginning step.

The text "Differential Geometry, Gauge Theories, and Gravity" is sufficient for the geometric part of the mathematical background of gauge theories in general, and even has a paragraph about lattice QCD! And applying this formalism to lattice QCD isn't "hard" (thus, a paragraph ;)). However, the "quantum" part of QCD is not covered by the classical gauge theories described in that text, and to really understand that you have to know QCD.

As for the physical intuition of what's going on, I doubt you will get much without a reasonable understanding of quantum field theories in general. You certainly can't get an understanding of these without losing rigor, since there is not yet a rigorous version of QFT (there are some very good books out there that do rigorously what we can do rigorously, but that is not much). Lattice QCD is on much better footing, but you do lose a lot of the intuition skipping straight to that.

So I would suggest going to the arxiv, and trying to find a good review article that makes you happy, and then check out its references when you get confused. But if you want to really develop a physical intuition for what's going on, I think that would be a lot of work.

Offhand, the book mentioned by Steve looks good, but if it is good for you or not depends on your background. And by math standards it does not look particularly mathematical.