One of the closer connections to geometric topology is likely from invariants of manifolds. The motivating reason for the development of topological modular forms was the Witten genus. The original version of the Witten genus associates power series invariants in $\mathbb{C}[[q]]$ to oriented manifolds, and it was argued that what it calculates on M is an $S^1$-equivariant index of a Dirac operator on the free loop space $Map(S^1,M)$. It is also an elliptic genus, which Ochanine describes much better than I could here.

This is supposed to have especially interesting behavior on certain manifolds. An orientation of a manifold is a lift of the structure of its tangent bundle from the orthogonal group $O(n)$ to the special orthogonal group $SO(n)$, which can be regarded as choosing data that exhibits triviality of the first Stiefel-Whitney class $w_1(M)$. A Spin manifold has its structure group further lifted to $Spin(n)$, trivializing $w_2(M)$. For Spin manifolds, the first Pontrjagin class $p_1(M)$ is canonically twice another class, which we sometimes call "$p_1(M)/2$"; a String manifold has a lift to the String group trivializing this class. Just as the $\hat A$-genus is supposed to take integer values on manifolds with a spin structure, it was argued by Witten that the Witten genus of a String manifold should take values in a certain subring: namely, power series in $\Bbb{Z}[[q]]$ which are modular forms. This is a very particular subring $MF_*$ isomorphic to $\Bbb{Z}[c_4,c_6,\Delta]/(c_4^3 - c_6^2 - 1728\Delta)$.

The development of the universal elliptic cohomology theory ${\cal Ell}$, its refinement at the primes $2$ and $3$ to topoogical modular forms $tmf$, and the so-called sigma orientation were initiated by the desire to prove these results. They produced a factorization of the Witten genus $MString_* \to \Bbb{C}[[q]]$ as follows:
$$
MString_* \to \pi_* tmf \to MF_* \subset \Bbb{C}[[q]]
$$
Moreover, the map $\pi_* tmf \to MF_*$ can be viewed as an edge morphism in a spectral sequence. There are also multiplicative structures in this story: the genus $MString_* \to \pi_* tmf$ preserves something a little stronger than the multiplicative structure, such as certain secondary products of String manifolds and geometric "power" constructions.

What does this refinement give us, purely from the point of view of manifold invariants?

The map $\pi_* tmf \to MF_*$ is a rational isomorphism, but not a surjection. As a result, there are certain values that the Witten genus does not take, just as the $\hat A$--genus of a Spin manifold of dimension congruent to 4 mod 8 must be an even integer (which implies Rokhlin's theorem). Some examples: $c_6$ is not in the image but $2c_6$ is, which forces the Witten genus of 12-dimensional String manifolds to have even integers in their power series expansion; similarly $\Delta$ is not in the image, but $24\Delta$ and $\Delta^{24}$ both are. (The full image takes more work to describe.)

The map $\pi_* tmf \to MF_*$ is also not an injection; there are many torsion classes and classes in odd degrees which are annihilated. These actually provide bordism invariants of String manifolds that aren't actually detected by the Witten genus, but are morally connected in some sense because they can be described cohomologically via universal congruences of elliptic genera. For example, the framed manifolds $S^1$ and $S^3$ are detected, and Mike Hopkins' ICM address that Drew linked to describes how a really surprising range of framed manifolds is detected perfectly by $\pi_* tmf$.

These results could be regarded as "the next version" of the same story for the relationship between the $\hat A$-genus and the Atiyah-Bott-Shapiro orientation for Spin manifolds. They suggest further stages. And the existence, the tools for construction, and the perspective they bring into the subject have been highly influential within homotopy theory, for entirely different reasons.

Hope this provides at least a little motivation.