Why are fusion categories interesting?
Fusion categories (over $\mathbb{C}$) are a natural generalization of finite groups and their behavior over $\mathbb{C}$. The complex representation theory of a finite group is a fusion category, but there are many others. In fact, you can think of a fusion category as a non-commutative, non-cocommutative generalization of a finite group. A finite-dimensional Hopf algebra is that too, but they don't have to be semisimple, while the semisimple ones give you many fusion categories, but again not by any means all of them.
Many of the basic results about the structure and representation theory of finite groups generalize, or seem like they could generalize, to fusion categories. This principle has been worked out to a very incomplete but interesting extent by Etingof and others. For instance there is an analogue of the theorem that the dimension of complex irrep of a finite group $G$ divides $|G|$. (Addendum: A qualified analogue, as Scott and Noah point out. If the category is braided, it is a strict analogue; otherwise it is an analogue of dividing $|G|^2$.) There are also semisimple Hopf algebras and other fusion categories that look a lot like $p$-groups.
You can think of the whole theory as a rebooted theory of finite groups. However, we are miles and miles away from any fusion category equivalent of the classification of finite simple groups. It is a struggle to make fusion categories that are not derived very closely from finite groups, or do not come from quantum groups at roots of unity. Only a few types of examples are known, and who knows what else is out there.
One enticing thing that does change is that dimensions of irreducible objects in a fusion category don't have to be integers. For instance, one of the simplest fusion categories is the Fibonacci category. It has two irreducible objects, the trivial one $I$ and the other object $F$. The dimension of $F$ is the golden ratio, as you can infer from the branching equation $F \otimes F \cong F \oplus I$. (But the dimensions are algebraic integers, and even cyclotomic algebraic integers. Hence divisibility is still a sensible question.)
You could also ask, why the semisimple case. As you learn in undergraduate or basic graduate representation theory, the semisimple representation theory of a finite group is much cleaner than the modular representation theory in positive characteristic.
And yes, you also get 3-manifold invariants and subfactors.
For references: Really Turaev and Viro's original paper, state sum invariants of 3-manifolds and 6j symbols, is pretty good. The generalization to spherical categories is due to Barrett and Westbury, Invariants of Piecewise-Linear 3-manifolds. And there is a discussion in Turaev's book.
A sketch: Recall that a basis-independent expression in tensor calculus has the structure of a graph with vertices labelled by tensors and edges labelled by vector spaces. A monoidal category allows the evaluation of similar expressions, except that the graph must be planar and acyclic. In a rigid pivotal category, there are good duals and the graph just needs to be planar. In a spherical category, left trace equals right trace, so a closed graph can be drawn on a sphere. If it is spherical, rigid, and semisimple, then you can use the graph of a tetrahedron to make a local interaction on the tetrahedra of a triangulated 3-manifold, and the result up to normalization is the Turaev-Viro 3-manifold invariant. (In this setting you should dualize the tetrahedra, so that a tensor morphism in the category is associated to a face of the tetrahedron.)
I am very far from an expert on the subject, but I have had to answer this question before. I will provide a few of my answers that seem to be omitted from the above discussion.
Fusion rings, or fusion rules, appear in certain physical "thought experiments" (though I am not aware that they have yet actually been observed). For instance "anyons" are defined by certain fusion rules,and if they existed would be useful in quantum computation. In the dogma of field theory, for these fusion rules to be physically meaningful, there needs to be a categorification of them, which would be a fusion category. This initiates a broad class of problems: given a fusion ring, decide whether or not it has a categorification, and if so, how many (up to equivalence). By Ocneanu's rigidity theorem, there are only finitely many categorifications of any given fusion ring, so it's possibly a "tame" problem to solve.
There is no hope (at present, per Greg's comment =]) to completely classify fusion categories in any sense, so far as I understand. The classification of fusion categories would encompass not only the classification of finite groups, but also of compact Lie groups (via the tilting module construction on the associated quantum group which yields a fusion category). So people classify fusion categories in small classes under the assumption that group theoretical categories (ones defined purely in terms of group theory: representations of groups, cohomology of groups, morita equivalences, etc.) is "easy" and they want to study the difference between the two contexts.
Greg mentioned that studying fusion categories is like studying semi-simple Hopf algebras, except that (a) there isn't necessarily a fiber functor to vector spaces, and (b) even if there does abstractly exist one, you don't choose one. If one admits the interest in studying Hopf algebras, then one has to admit the interest in fusion categories as a sort of "basis free" version. A direct application to finite groups is pinning down the precise relation between the groups D_8 and the quaternions. They are obviously not isomorphic; however their group rings are morita equivalent as rings (since they have the same number of irreducibles). Their irreps even have the same dimensions, so one can ask if their fusion categories are equivalent as fusion categories (they are not in this case, but there are some non-isomorphic groups which are so-called "isocategorical" meaning that not only are their group rings isomorphic, but the Hopf algebras are twist equivalent as Hopf algebras. The most sensible way to prove this sort of statement is through fusion categories.
For me, I am a fairly concrete-minded person, but someone who nevertheless tries to understand modern algebraic geometry, algebraic topology and category theory as best I can. Fusion categories have been a fantastic discovery for me, because they are in many ways homotopy theoretic/higher category-type constructions, but they are about as simple as one can get (because you basically have constrained the 1-morphisms as much as possible by the semi-simplicity assumption, and just focus on the higher morphisms). So for instance the first 2-groupoid I was ever able to understand in completely concrete terms arises in a paper of Etingof Nikshych and Ostrik about fusion categories. As such they can be viewed as a kindergarten of higher categories.
By the way, there is also some interest in "finite tensor categories" which are not semi-simple but satisfy the other finiteness conditions of fusion categories. (so you have finitely many simple objects, and you posit that every object is a finite-length extension of the simples). There's actually a great deal of the theory from fusion categories which generalizes here. So far as I can tell, the only obstacle in developing this notion more completely is that no one has had time to do it yet.
I also wrote a sequence of blog posts explaining the Turaev-Viro construction from the point of view of planar algebras. It has pretty pictures and might be relevant.
TQFTs via Planar Algebras I
TQFTs via Planar Algebras II
TQFTs via Planar Algebras III