The problem with "why"
"Why" is generally an unfortunate question. It is often very hard or impossible to answer. The reasoning goes like this: if you ask "why" a (reasonably complex) thing is like it is, any meaningful answer usually breaks the issue down into subcomponents. Then, you can and need to ask "why" for each of those. This goes on and on ad nauseam. Either you end up with more and more ever more complex answers and questions; or you eventually lose the semantics along the convoluted path, or you end up with subquestions that simply have no meaningful "why" answer at all.
A popular example would be children asking why the sky is blue. With a "nerd" child, you can spend a lot of time on that. It's not uninteresting, and not futile, and being inquisitive is great, but it's often frustrating.
There is a nice video from Feynman himself explaining this: https://www.youtube.com/watch?v=36GT2zI8lVA
Why do we use rings, groups, fields in cryptography
We use these structures in cryptography because in modern times, cryptography inevitably works on natural numbers as their base domain, and nothing else. Why is that the case? Because in our current computer architecture, all data is represented as natural numbers sooner or later, and therefore it only makes sense to focus all our efforts on this common denominator, instead of having different cryptography for images, text, sound etc..
Rings, groups, fields and other structures are literally what makes up numbers (or rather the semantics on numbers). There is no magic to this. Every child is working with these structures even when learning to count to 100, they just don't know it yet. It's not like mathematicians saw all the problems of cryptography, tried 10 alternatives, until someone said "hey, let's take rings", but whenever you are working with numbers, you inevitable end up with some of these structures.
It's not like we invented all these structures (people have been multiplying and adding long before these terms or the strict understanding of their structure were around); we just use them to label some very restrictive sets with their operations. Other labels or definitions could probably have been arrived at.
A direct example is the RSA public key algorithm. It is a conceptionally simple set of operations in modular arithmetic (i.e. $\mathbb N_n$ with $n$ being a very large integer with the property $n = pq$, with $p$ and $q$ being large primes). When working with these things, the only way we know how is to work with operators (multiplication...) which form a group structure. Ignoring key generation, the actual encryption and decryption is a single operation: exponentiation in modular algebra. Understanding why it works is easy even if someone only had a few first lessons in algebra, to define the terms. Hence we are using groups here (or rather some practical proofs from group theory) because they are the correct tool, they make it easy to reason about the algorithm and prove the correctness, and very easy to implement.
As a side point, not directly related to the question: Here, it just so happens that this operation also is very simple for a computer to perform. The inverse (which would be equivalent to cracking the code) is very hard. "Why" that is (on a philosophical level) is impossible to tell, except we didn't find a way to do yet. Nobody knows if it is fundamentally impossible to ever find an efficient algorithm to invert this.