If you have two variables, and changing one results in a the other changing as well, we say that those variables are proportional. If that change is just multiplying by a constant, then we call that constant the constant of proportionality.
If we look at the equation y = 5/8x, then we know x and y are proportional, because changing one results in the other changing as well. Furthermore, we can see that if we change x, then we are changing y by that amount multiplied by 5/8. For example, it's easy to see that if x=0, then y = 5/8*0 = 0. Let's see what happens to y when we add n to x:
y = 5/8*(0+n) = 5/8n = 0+5/8n
So, when we increased x by n, we increased y by 5/8n. We can use a similar process to show that it doesn't matter where we start. A change of n in x will result in a change of 5/8n in y. This matches our description of a constant of proportionality.
You're also right about it being the slope. In a line, the slope is just another way of expressing the change in y with relation to the change in x. That is often the first way students are introduced to it:
"slope" = "rise"/"run" = ("change in "y)/("change in "x)
A very fast way to find the slope of a line given the equation is to put it into point-slope form.
If a line has the equation y = mx + b, where m and b are constants, then m is the slope of the line and b is the y-intercept (that is, the line passes through the point (0, b)).
Note that the given equation is actually in that form already, with m = 5/8 and b = 0.
