quantum topology - Why is the volume conjecture important?

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 $langle q_i|e^{-Ht}|q_frangle$ 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!