How do the evanescent waves contribute/cause a diffraction limit?

Suppose our object is described by the function $g(x,y,z=0)$. We can write this function as a superposition of plane waves (Fourier decomposition), $$ g_{object}(x,y,z=0) = iint df_x df_y ;G(f_x, f_y,z=0)e^{-2pi i (f_x x + f_y y)} $$ where $f_x$ and $f_y$ are the spatial frequencies. Hence, the object consists of many different spatial frequencies.

Now, if we use the fact that any imaging system (=lenses) possesses a finite resolution, we conclude that after propagating the distance $z$ some spatial frequencies, which were present in the object and thus in $G(f_x, f_y,0)$, have vanished from the image. Hence, the propagation of the light changes the function $G(f_x, f_y,z)$ -- which is why we included the variable $z$. In fact, the free space propagator is given by $$G(f_x, f_y,z) =G(f_x, f_y,0) cdot exp{left( ikzsqrt{1 - lambda^2(f_x^2 + f_y^2)} right)} $$ where you see that for large enough spatial frequencies the term under the square root becomes negative. These are the evanescent waves.

A full discussion can be found in the textbook from Goodman "Fourier Optics".