Don't need stable. It's the stack of all G bundles on a curve. The right hand side is the derived category of local systems, which are vector bundles with flat connection (G-bundles, for the Langlands dual of G in this equation). As for background, that depends on how deep an understanding you want.

For a lot of the positive characteristic stuff, you want to look at stuff by Frenkel and Gaitsgory, is my understanding.

In the complex case, there's a bit of other stuff. I've got to, of course, recommend the work of my advisor, as well as the work of Witten which is closely related. To really get this, you need to have a bit of an understanding of the Hitchin system on Higgs bundles (though not by that name, they're used extensively in this paper).

As for why it's called Geometric Langlands, that'd be because it's, essentially, a geometric formulation related to the classical Langlands conjecture in number theory, which I know fairly little about, except that it involves automorphic forms. I recommend asking your local number theorist for some help, though I do believe that it is explained in the book Introduction to the Langlands Program by quite a few authors, which includes Gaitsgory explaining roughly what Geometric Langlands is and how it all fits together.

Hope that helps.