Let me give you one reason.
As you may know moduli spaces of curves are important objects. The homology of moduli spaces stabilizes when the genus increases (by Harer's stability theorem) and is known thanks to the solution to Mumford's conjecture. In the unstable range, computations become very difficult C.-F. Bodigheimer has done some computations (see also V. Godin's paper to Math. Ann.). The solution of Mumford's conjecture (by Galatius-Madsen-Tillmann-Weiss) relies on Segal's foundationnal work on field theories because of the heavy use of topological categories of cobordism. This is a beautiful example of how Segal's ideas together with homotopy theoretic techniques can be applied to a problem in algebraic geometry.

Studying topological conformal field theories is studying the space of representations of the prop of singular chains of these moduli spaces. One hope is to be able to say something about the homology of this prop through the eyes of its representations.

Let me give you an example. If you look at string topology, we are able to build operations by considering the action of spaces of diagrams (which are combinatorial models of moduli spaces they are related to Kontsevich's graph homology) on the singular chains of the free loop space of a closed, oriented manifold $M$. Singular chains of loop spaces or at least Hochschild cochains of the cochains of $M$ give an example of a TCFT (with some boundary conditions), you can view it as a consequence of Costello's work (or of the cobordism hypothesis).

Imagine that you have a geometric cycle in your space of diagrams and that you are able to prove that its action is not homologicaly trivial on a free loop space then you have created a homology class in the homology of the corresponding moduli space.
References here are : Papers of Chas and Sullivan, Cohen and Godin, Godin, Kupers, Poirier, Wahl.....(I am sure I have forgotten plenty of people). Nathalie Wahl has a preprint in preparation available on her homepage toward this direction.

**Edit :** Nathalie Wahl's preprint is now published:
"Universal operations in Hochschild homology", J. Reine Angew. Math. 720 (2016), 81-127. arXiv:1212.6498, doi:10.1515/crelle-2014-0037

In section 4 of this paper she exhibits a family of non-trivial string topology operations $t_g$. These operations are unstable homology classes of moduli spaces of genus $g$ riemann surfaces with $1$ incoming and 1 outgoing boundary components.