"1 x 1" means whatever you want it to mean. There is nothing that says the operation "x" should be multiplication. There is nothing that says that "1" should even be a number. I could, just to be a regular A-hole, say that "1" is shorthand for a set of elements, and "x" is just a nice way of pairing "1" with itself, and I can call this a domain for a function I am about to define.
However, if we are being totally serious here, "x" means the operation of multiplication and "1" is a commonly understood concept referring to the idea of a single thing by itself. Multiplication is a short-cut for lengthy addition problems.
So 3x4 is a short cut for 3 + 3 + 3 + 3, or equivalently, 4 + 4 + 4. That's pretty much it. How many times do I add up 3? How many times to I add up 4?
So when you ask what is "1 x 1", you are really asking How many times do I add up 1? Well, you add up 1 just 1 time. That gives you 1.
