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][1] and [Gaitsgory][2], 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][3], as well as the work of [Witten][4] 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 the paper _[Spectral curves and the generalised theta divisor][5]_, Crelle 1989).

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][6]_ by quite a few authors, which includes Gaitsgory explaining roughly what Geometric Langlands is and how it all fits together.

Hope that helps.


  [1]: https://arxiv.org/search/advanced?advanced=&terms-0-operator=AND&terms-0-term=Frenkel&terms-0-field=author&terms-1-operator=AND&terms-1-term=Langlands&terms-1-field=all&classification-mathematics=y&classification-physics_archives=all&classification-include_cross_list=include&date-filter_by=all_dates&date-year=&date-from_date=&date-to_date=&date-date_type=submitted_date&abstracts=show&size=50&order=-announced_date_first
  [2]: https://arxiv.org/search/advanced?advanced=&terms-0-operator=AND&terms-0-term=gaitsgory&terms-0-field=author&terms-1-operator=AND&terms-1-term=Langlands&terms-1-field=all&classification-mathematics=y&classification-physics_archives=all&classification-include_cross_list=include&date-filter_by=all_dates&date-year=&date-from_date=&date-to_date=&date-date_type=submitted_date&abstracts=show&size=50&order=-announced_date_first
  [3]: https://arxiv.org/search/advanced?advanced=&terms-0-operator=AND&terms-0-term=Donagi&terms-0-field=author&terms-1-operator=AND&terms-1-term=Langlands&terms-1-field=all&classification-mathematics=y&classification-physics_archives=all&classification-include_cross_list=include&date-filter_by=all_dates&date-year=&date-from_date=&date-to_date=&date-date_type=submitted_date&abstracts=show&size=50&order=-announced_date_first
  [4]: https://arxiv.org/search/advanced?advanced=&terms-0-operator=AND&terms-0-term=witten&terms-0-field=author&terms-1-operator=AND&terms-1-term=Langlands&terms-1-field=all&classification-mathematics=y&classification-physics_archives=all&classification-include_cross_list=include&date-filter_by=all_dates&date-year=&date-from_date=&date-to_date=&date-date_type=submitted_date&abstracts=show&size=50&order=-announced_date_first

  [5]: http://math.unice.fr/~beauvill/pubs/bnr.pdf
  [6]: https://doi.org/10.1007/978-0-8176-8226-2