Wikipedia's description of the Casson invariant gives the first important reason to study it. As an invariant that comes from the $\text{SU}(2)$ representation variety of $\pi_1(M)$, it reveals in particular that $\pi_1(M)$ is non-zero. At the time, before Perelman's proof of the Poincaré conjecture and geometrization, there was a lot of mystery about potential counterexamples to the Poincaré conjecture. For instance, one speculation was that the so-called $\mu$ invariant could reveal a counterexample. Since the Casson invariant lifts the $\mu$ invariant, and since it proves that $\pi_1(M)$ is non-trivial when it is non-zero, it is one way to see that the $\mu$ invariant can never certify a counterexample to the Poincaré conjecture. (Of course, no we know that there are no counterexamples.)
A second fundamental reason to study the Casson invariant is that it is the only finite-type invariant of homology spheres of degree 1. Many interesting 3-manifold invariants are finite-type, or (conjecturally) carry the same information as a sequence of finite-type invariants. This is known more rigorously at the level of knots; for instance, the derivatives of the Alexander polynomial, the Jones polynomial, and many other polynomials at $1$ are all finite-type invariants. At the level of knots, the second derivative of the Alexander polynomial, $\Delta''_K(1)$, is known to be the only non-trivial finite-type invariant of degree 2, and there is nothing in degree 1. So it means that this invariant appears over and over again as part of the information of many other invariants; there are many different definitions of the same $\Delta''_K(1)$. The same thing should happen to the Casson invariant, and indeed there are already two very different-looking types of definitions: (1) Casson's definition; (2) either the first LMO invariant or the first configuration-space integral invariant.
A third fundamental reason is that Casson invariant has an important categorification, Floer homology, which is the objects in the theory whose morphisms come from Donaldson theory. One wrinkle of this construction is that it is only a categorification of one of the definitions of Casson's invariant, Casson's definition. If Casson's invariant has many definitions, then it might (for all I know) have many different categorifications.
If your question is meant in the narrow sense of what topology you can prove with the Casson invariant, then you can definitely prove some things but only (so far) a limited amount. However, if you are interested in quantum topological invariants in their own right, and not just as a tool for pre-quantum topology problems, then the Casson invariant is important because it is a highly non-trivial invariant that you encounter early and often.
Best Answer
I'll give a shot at an answer. The relevant dimensions are of the form $2^j-2$. For $j\leq 4$, it is easy and classical that we can construct manifolds of Kervaire invariant one. The problem was ``reduced'' from differential topology to pure stable homotopy theory by Browder in 1969. Direct calculational methods in homotopy theory were used by Barratt, Jones, and Mahowald to construct a cell complex that can be used to solve the homotopy theory problem and prove that such manifolds also exist in dimensions 30 and 62. I believe a construction of such a manifold has been worked out in dimension 30, but that has certainly not been done in dimension 62. Periodicity phenomena play a huge role in modern stable homotopy theory, and a crucial feature of the Hill, Hopkins, Ravenel proof is a periodicity of order $2^8 = 256$. That enables them to solve the stable homotopy problem and prove there is no manifold of Kervaire invariant one for $j\geq 8$. The reasons $j=7$ is so hard are several. Nobody has a really good reason for guessing which way the answer will go. There is no reason to expect a relevant periodicity of order $2^7$. Direct calculation of the Adams spectral sequence through dimension $126$ is just plain hard: the calculations blow up. There is a chance that the methodology of Barratt, Jones, and Mahowald might extend to prove existence (if that is how the answer turns out!), but it will probably be much harder to prove nonexistence (if that is the answer).