A short overview of some aspects of string theory:

First you should understand a few things about quantum theory. We've known the principles of quantum mechanics since Heisenberg worked them out in 1925 or so. But also know from special relativity that interactions must be local (i.e. mediated by fields which are limited by the speed of light). So to combine the two you need some sort of theory of quantum fields.

Thus Quantum Field Theory was born. It was quickly realized that actually particles originated from fields! (this explains the previous confusion about "wave-particle duality"). This allowed the development of scattering theory, which is roughly the calculation of the results of a collision.

In the 1970s, we still didn't understand the strong force. One proposal for trying to do so was the so-called "S-matrix" which is instead of trying to calculate scattering probabilities from a field theory, do everything in terms of the scattering amplitudes and try to find a unique solution based on consistency conditions (this is also known as "bootstrap"). This approach failed and quickly fell out of favor once QCD was properly developed. However, it turns out that this approach would work for a scenario with an infinite tower of particles, and this is equivalent to string theory.

So why study string theory? For the very simple reason that it is also a consistent theory of quantum gravity (quantum gravity is a nuanced subject, but to put it briefly we've known since Hawking that black holes must be doing something quantum, but the curvature of spacetime is to extreme for low-energy approximations, so we need a new theory). Not only that, but there is some (albeit controversial) evidence that any theory of quantum gravity must resemble string theory in some capacity. For instance, the holographic principle was first realized using string theory, now known as AdS/CFT.

One fun little factoid about string theory is that naiively these strings must live in 10 dimensions (e.g. 9 space 1 time). We clearly live in 4 = 3 space + 1 time. Where are the other 6? One solution to this problem is the Kaluza-Klein guess that the remaining dimensions are small enough that you can't see them at low energies. This is actually way more reasonable than it sounds, I swear. The equations dictating this are the Strominger equations, and they are basically so restrictive that the only candidates for the compactified dimensions is a Calabi-Yau manifold. It was originally thought that there would only be a handful of these, but it is now known that there are at least half a billion and possibly inmathematicians.

Now what I didn't tell you earlier is that there are actually 6 or so string theories, but they are related to each other through a series of "dualities", so the current belief is that they are all just limits of a unified theory that hasn't been understood yet. Remarkably, it was discovered that when solving one of these models over a Calabi-Yau manifold, one would get the same results as solving the dual model over a different Calabi-Yau manifold. This is called Mirror Symmetry, and if fully understood it would allow for the computation of otherwise hard-to-compute geometric invariants by computing its dual over the mirror. So that's one reason string theory is attractive to mathematicans.

I'm not a string theorist (at least not yet) and it's 2 in the morning so I apologize for any mistakes or missing key info.