I think there are really three issues buried in your question.
1. What's Simon's Problem?
2. What's a plain english description of the function involved in Simon's Problem?
3. Why isn't this problem a cryptographic primitive?
I can't really speak to 3 as cryptography is not my area. I can take a crack at 1 and 2 for you though. The distinction between 1 and 2 is, I think, important.
<PERSON>'s problem is one of discovering the parameter to a function given a black-box to that function and some basic information about the function. So, an algorithm to solve this problem takes as its input a black-box to a function and gives a output a binary string. Our goal is to reduce the number of queries to that black box. I think people tend to miss this point and focus on the internals of the function itself.
All we know about the function $f$ is that our black-box can compute it for us in constant, $O(1)$, time, and that if $f(x) = f(y)$, then either $x=y$ or $x \oplus y = s$. 603
We want to know the value of $s$. A brute force, classical computing solution might give you a better feel for the problem. Here it is:
1. Feed all $n$ bit binary strings to the black-box and save each input-output pair to a table.
2. If no two inputs produce the same output then it must be the case that the parameter $s$ is $n$ bit $0$. Otherwise, two inputs will produce the same output and $s$ is the result of XOR'ing those inputs together.
In the worst case, we have to query all $2^n$ inputs leading to $O(2^n)$ query complexity.
Querying the black-box is at the heart of <PERSON>'s problem. We need not concern ourselves with how to construct the black-box, i.e. how to compute $f$, as that is not what the problem is about. I don't believe that <PERSON>'s problem is practically useful beyond the fact that it helps establish some complexity results about Quantum Computers. It's main significance is that it was an inspiration for <PERSON> factoring algorithm. 603