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I've forgotten what the scientific term is for explaining a complex theory via a simpler one, but this is about as extreme an example as you can get.
To visualise mathematics in its most basic and intuitive form we must do away with number systems and use dots instead of numerals.
See attached image →
I will make more images later but for now I will just describe what follows.

Now let's imagine these dots as tiny particles whizzing about and occasionally colliding — let's not worry about how these particles came to be for now. We've already established that 1·1=1, so in a world where only two particles exist surely they will eventually collide and become a single particle? So how can we ever get more than two particles in this system? Hundreds of particles could spontaneously form at once, but even then it's a short matter of time before they all annihilate back to 1.
The only way to have a stable system of >1 particles is if multiple particles can form stable groups — imagine a group of dots spinning around each other but never colliding.

So what happens when two groups collide? All dots collide between groups simultaneously. So what actually happens when these dots collide?
This is mind blowing part 1: When the dots collide they break apart into smaller dots, some collide with each other producing even smaller dots, and others form stable groups on the same scale. So a single collision between two dots is an infinite cascade of collisions between smaller and smaller dots.
Mind blowing part 2: Groups of smaller dots can join with other groups to make larger groups. Therefore the two dots we started with were both in fact groups of smaller dots, and each of those was a group of smaller dots, getting smaller and smaller and smaller to infinity!

So what is actually happening during multiplication? The collision, which we now know is just one super-group interacting with another super-group, causes a change in forces that breaks apart the super groups and flings them into a field filled with smaller groups, and probability determines how many of these groups combine into new super-groups.

So how does this relate to 1×1=1? The two groups collide and break into smaller groups, probability takes over here and dictates that enough groups will recombine to produce 1 super-group.

Now it is clear that talking about numbers as discrete values is ridiculous. What is a 1 if it is comprised from infinitely many parts? It is simple a value on a particular scale.
Nothing is spontaneously created because it existed before on a smaller scale.

Thinking Points: Why do interactions between groups always produce the same outputs? Why is there a limit to how many smaller groups can be combined during this reaction?