Question

I don't get this at all

Answer 1

Δy/Δx is nothing but the slope and slope = rise/run .

Look up in the FIRST graph; From the lower point on the line draw a // to x-axis and from the higher point draw a // to y-axis. Note the point of intersection, their coordinates are (6, 3) [ x=6 = run and y= 3 = rise)
Or Δy = rise = 6    and Δx = run = 3, hence Δy/Δx  = 6/3 = 1/2 = slope.

Do the same thing with the 2nd graph:
Δy = 3 and  Δx = - 2 (you are counting from right to left), then 

Δy/Δx = 3/(-2)  and Δy/Δx = -3/2