Maybe I'm just being slow, but:
Why is the slope formula $\displaystyle \frac {\Delta y}{\Delta x} $?
More specifically, why do we divide the change in y by the change in x to get the slope, or incline between two points?
It can be a very hard question, depending on what you mean. That ratio is basically a quite interesting property of a curve, and has been named slope. You simply want to know how much y increases (or decreases) in proportion to how much x increases. That's as simple.
1) It's a definition. It doesn't have to be any trickier than that.
2) It is a useful definition that provides a basis for comparison. Example:
If I tell you one line increases vertically by 12 and another line increases vertically by 5, what can you say about the two increases? Is one line rising faster than the other?
I await your response.