I can't get my head around how dividing the differences of two points gives us the inclination of a line. I do understand that the slope will show how much or little a change in $X$ will affect the change in $Y$ but I can't figure out why we find the slope by division. Does division somehow mean average out? If so why?
Thanks in advance.
Welcome to MSE!
Let's work with lines through the origin, with the equation $y=mx$. The $+b$ does nothing but shift the line up and down, which does not interact with the slope at all.
As you've noticed, the slope $m$ in the formula $y=mx$ tells you how much the $y$ coordinate changes compared to the $x$ coordinate. For instance, if we change our $x$ coordinate by $1$ (say by moving from $x=3$ to $x=4$) then our $y$ coordinate changes by $m$ (it moves from $y=3m$ to $y=4m$).
In particular, if we know the slope $m$ and the change in $x$ (call it $\Delta x$) then we can compute the change in $y$ (call it $\Delta y$) by $\Delta y = m \Delta x$ (do you see why?).
But we're interested in going the other direction. If we know how $y$ changes with respect to $x$, we want to compute the slope. But since division undoes multiplication, we can solve the above equation for $m$ to find
$$m = \frac{\Delta y}{\Delta x}$$
as you surely expected.
As a quick exercise, what happens if we use the general formula $y=mx+b$ instead of the simplified $y=mx$? Do you see why the $+b$ doesn't change any of our computations? It will have something to do with the fact that we're working with changes in $x$ and $y$, whereas the $+b$ doesn't change!
I hope this helps ^_^