This slide gives a sketch of an explanation for why attribute/w is linear in 2D screen coordinates. This is important because the rasterizer can simply set up a linear equation per attribute f, that allows it to compute f/w for any screen sample point (x,y).
Note that for simplicity, the perspective projection matrix on this slide is a simple one, it's not the exact matrix that you would use to convert a point into OpenGL's normalized device coordinates, which you can see here. But notice that using a real OpenGL perspective projection matrix would only modify things in that the coefficients in from of x_2d and y_2d in the final equation would be different. The main point, that f/w is affine function of x_2d and y_2d still holds.