We use y-intercept and slope because those are the two things you need to graph a line. The slope intercept form of a line is y=mx+b, where m is the slope and b is the y-intercept.
For example, if you know the y-intercept is 3 and the slope is 2, the equation of the line is y=2x+3.
To graph: you already know one point on the line, the y-intercept, which is 3. So the point is (0,3).
You know the slope is 2, so from (0,3) go up 2 and right 1, since slope is change in y over change in x (2/1). So the next point would be (1,5)
Now with the two points (0,3) and (1,5) you connect them to form a line (since two points form a line).