Why is the volume conjecture important? The volume conjecture, a formula relating hyperbolic volume of a knot complement with the semiclassical limit of a family of coloured Jones polynomials, is widely considered the biggest open problem in quantum topology. It is one of a large family of conjectures and research programmes which have to do with  detecting classical geometry with semiclassical limits.
Embarassing as it is to say in public, I only very partially understand why people care so much about such conjectures.

What fantastic consequences would there be for low dimensional topology if the volume conjecture were proven tomorrow? What if all the related conjectures were proven too? How would it improve our understanding of classical topology? More broadly, how would it advance mathematics?

 A: I don't know any consequences for low-dimensional topology. My impression is that it would indicate how powerful TQFT invariants are at distinguishing 3-manifolds or knots. For example, the volume conjecture (or maybe variants by the Murakamis) would imply that the colored Jones polynomials distinguish knots from the unknot. This corollary is also claimed by Jorgen Andersen. There are only finitely many hyperbolic knots of a given volume, so the volume conjecture would imply that the colored Jones polynomials are close to distinguishing hyperbolic knots.  
A: Someone else will have to discuss the applications in topology, but I can point out at least one reason the volume conjecture is interesting.  
It's often said that no one knows how to define the functional integral for Chern-Simons theory.  This isn't literally true.  The Reshetikhin-Turaev construction can be interpreted -- tautologically -- as defining a volume measure on a certain space of functionals.  (This is just like in quantum mechanics, where one interprets the kernel $\langle q_i|e^{-Ht}|q_f\rangle$ as the volume of the space of paths $\phi: [0,t] \to \mathbb{R}$ which begin at $q_i$ and end at $q_f$.)  What we don't know how to do is define the path integral measure as a continuum limit of regularized integrals that look like $\frac{1}{Z}e^{iCS(A)}dA$.
The volume conjecture (in particular the version where log of the Jones polynomial looks like vol(3-manifold) plus i times the Chern-Simons functional)  tells us that the tautological measure you get from Reshetikhin-Turaev actually has something to do with the Chern-Simons action!
