OK, this is a very broad question so I'll be telegraphic.
There is a sequence of increasingly detailed conjectures going by the name GL -- it's really a "program" (harmonic analysis of $\mathcal{D}$-modules on moduli of bundles) rather than a conjecture -- and only the first of this sequence has been proved (and only for $GL_n$), but I don't want to get into this.
There are several kinds of reasons you might want to study geometric Langlands:
direct consequences. One application is Gaitsgory's proof of de Jong's conjecture (arXiv:math/0402184). If you prove the ramified geometric Langlands for $GL_n$, you will recover L. Lafforgue's results (Langlands for function fields), which have lots of consequences (enumerated eg I think in his Fields medal description), which I won't enumerate. (well really you'd need to prove them "well" to get the motivic consequences..)
In fact you'll recover much more (like independence of $l$ results). To me though this is the least convincing motivation..
Original motivation: by understanding the function field version of Langlands you can hope to learn a lot about the Langlands program, working in a much easier setting where you have a chance to go much further. In particular the GLP (the version over $\mathbb{C}$) has a LOT more structure than the Langlands program -- ie things are MUCH nicer, there are much stronger and cleaner results you can hope to prove, and hope to use this to gain insight into underlying patterns.
By far the greatest example of this is Ngo's proof of the Fundamental Lemma --- he doesn't use GLP per se, but rather the geometry of the Hitchin system, which is one of the key geometric ingredients discovered through the GLP. To me this already makes the whole endeavor worthwhile..
- Relations with physics. Once you're over $\mathbb{C}$, you (by which I mean Beilinson-Drinfeld and Kapustin-Witten) discover lots of deep relations with (at least seemingly) different problems in physics.
a. The first is the theory of integrable systems -- many classical integrable systems fit into the Hitchin system framework, and geometric Langlands gives you a very powerful tool to study the corresponding quantum integrable systems. In fact you (namely BD) can motivate the entire GLP as a way to fully solve a collection of quantum integrable systems. This has has lots of applications in the subject (eg see Frenkel's reviews on the Gaudin system, papers on Calogero-Moser systems etc).
b. The second is conformal field theory (again BD) --- they develop CFT (conformal, not class, field theory!) very far towards the goal of understanding GLP, leading to deep insights in both directions (and a strategy now by Gaitsgory-Lurie to solve the strongest form of GLP).
c. The third is four-dimensional gauge theory (KW). To me the best way to motivate geometric Langlands is as an aspect of electric-magnetic duality in 4d SUSY gauge theory. This ties
in GLP to many of the hottest current topics in string theory/gauge theory (including Dijkgraaf-Vafa theory, wall crossing/Donaldson-Thomas theory, study of M5 branes, yadda yadda yadda)...
- Finally GLP is deeply tied to a host of questions in representation theory, of loop algebras, quantum groups, algebraic groups over finite fields etc. The amazing work of Bezrukavnikov proving a host of fundamental conjectures of Lusztig is based on GLP ideas (and can be thought of as part of the local GLP). (my personal research program with Nadler is to use the same ideas to understand reps of real semisimple Lie groups). This kind of motivation is secretly behind much of the work of BD --- the starting point for all of it is the Beilinson-Bernstein description of reps as $\mathcal{D}$-modules.
There's more but this is already turning into a blog post so I should stop.
Let me try to answer. [FGV] is only about unramified representations of the Galois group
but they prove a stronger fact in this case (existence of certain "automorphic sheaf"). Lafforgue's result doesn't follow from there for several reasons:
a) Formally [FGV] use Lafforgue, but this was actually taken care of by a later paper of Gaitsgory ("On the vanishing conjecture..."). So that is really not a problem now.
b) Extending [FGV] to the ramified case is not trivial. I actually suspect that it can be done using the thesis of Jochen Heinloth but this has never been done (even the formulation is not completely clear in the ramified case)
c) In the unramified case what follows immediately from [FGV] is that you can attach a cuspidal automorphic form to a Galois representation. It is not obvious to me that the converse statement follows (Lafforgue's argument actually goes in the opppsite direction:
he proves that a cuspidal automorphic form corresponds to a Galois representation and then the converse statement follows immediately from the converse theorem of Piatetski-Shapiro et. al.
and from the fact that you know everything about Galois L-functions in the functional field case).
Best Answer
EDIT Few days ago a survey by A. Parshin appeared in arxiv. I think it is the best place to look on the higher-dimensional Langlands. From the abstract:
Actually similar question has been asked. Higher-dimensional Langlands is something very intriguing. I did not follow recent advances, but let me mention some part which I know about.
Let us start with NON-geometric local case. Langlands correspondence is roughly speaking "bijection" between representations of Galois group and representations of GL(Local Field).
Abelian case (class field theory) is bijection between characters of Galois and characters Local field, if dualize we get Galois/[Galois,Galois] = (Local Field)^*.
One of the first questions to ask - whether it is possible to generalize local class field theory to higher dimensions ?
The main idea by A. Parshin is that in n-dimensions one should consider Milnor's (n)-th K-group of local field instead of (Local Field)^. In particular for n=1 K_1^Milnor(Field)=Field^. (By the way the definition of higher dimensional local field should also be given). Parshin also found higher analogs of various symbols and proved higher analogs of reciprocity laws.
To the best of my knowledge there were no further developments in the field before Kapranov's paper "Analogies between the Langlands correspondence and topological quantum field theory" . In that paper he gave certain vision what higher dimensional Langlands might be.
His idea (quite amazing) that "representations" should be substituted by "k-representations" (i.e. representations in higher categories), (e.g. for surfaces we should consider 2-representations). So in n-th dimensions k-representations of dimension r of Galois group should correspond to (n-k)-representations of GL_r(n-Local Field)
In particular abelian version will correspond to Parshin's higher dimensional class field theory, since Milnor's K-groups correspond higher-representations.
Now about the geometric version. It is quite unclear for me. In 1-dimension, we can think of flat connections as analogs of Galois representations. It seems in higher dimensions we should consider gerbes with flat connections as analogs of higher representations of Galois group, whatever it means...
The geometric substitute for moduli space of vector bundles and Hecke-eigen sheaves is not clear for me. The reason is that in 1-dimension moduli space of vector bundles arise as standard coset G_{out}\G(k((z))/G_{in}, however in higher dimensions I do not see how the group G( k((z,u)) ) may have finite-dimensional quotient. So I do not see something finite-dimensional where "n-Hecke eigensheaf" may live.
Well, it is probably just my own problem. I have speculated around these things in http://arxiv.org/abs/hep-th/0604128, but I am afraid it is very unclear...
Last year there appeared a paper Unramified two-dimensional Langlands correspondence, which probably is the last mile achievement in the question. Unfortunately I have no time to follow these developments.