Point slope form: y - y1 = m(x - x1)

I realize this is a manipulation of the slope equation: m = rise/run, but I don't understand what is happening or why it works. The difference in your y values is equal to the difference of your x values multiplied by the slope. I'm having trouble seeing this. I don't exactly understand what is going on here. How and why?

I came to ask if someone could please clear this confusion up for me? I would really appreciate the help in understanding this!

Please go slow with your explanation so I understand. x)