Trouble in proving the simplest case of central limit theorem from a convolution viewpoint?

Question

I have once viewed an stanford video, which proves the CLT from a convolution viewpoint rather than using the moment generating function and characteristic function etc. I felt the convolution viewpoint more intuitive. But I was not able to grasp the proof. Now I have no access to that stanford video or any text book. So I searched internet, but still can't grasp the proof. The professor starts with defining a random variable $z = x + y$, where x and y are other i.i.d RVs. How can I prove from this to show $f_Z(z) = f_X(x) \ast f_Y(y)$ where $\ast$ stands for convolution and the functions stand for probability density functions.