Mathematics studies structures and is the language in which we can talk about these beings in a rather precise way (define them, express their properties, behaviour,...).
If we take a simplistic view of the universe, we could say that it is made of lego bits. These bits happen to fit with each other or can create new bits when something happens to them. These can be put together to form structures (or maybe they are structures already but where do we start?), and we have an increase of complexity that after some time gave rise to things like us.
There may be universes where these bits can not be put together, or the structures they form are more unstable than in ours. Since structures can be talked about with Mathematics, thus it appears as Mathematics can explain our universe. I say appears because it is a byproduct of the original structures and because OUR Mathematics does not really study all possible structures.
The fact is that we do a very human type of Mathematics. If you want me to put a number on it, I would say that with today's Mathematics we study less than 1% of what is out there. We are mostly interested in "nice" structures (that appeal to our sense of beauty). For example, a Ring in which lets say multiplication is associative for all its elements (otherwise it would not be called a Ring) is nicer than a "Ring" in which multiplication with all elements but one (or two, three...) is associative.
Why is it nicer you ask? Well, for starters it is easier to define, easier to find examples to work with and behaviour is more predictable. Just try to find an example of the other "Ring" or try to prove that such structure does not exist (there you have a life work).
So far we have been lucky that our "nice" Mathematics can explain so much of the universe. But if we find things we can not explain, then it becomes very interesting: Is it because we have not developed Mathematics enough? Or is it because there are structures that can not be explained with Mathematics at all? In the later case, are there universes out there that work on such structures? Is ours one of them and we have not realized it?
There is surely a limit to what our "nice" Mathematics can explain. What do we do when we get there? Well, we can try to evolve faster and become something better at math :)