The pdf of the sum of two random variables is the result of the convolution of the two initial pdf's, which can be seen as a low-pass filtering of the first distribution with the second, mirrored (in other terms, a sliding weighted average). The mirroring comes from the fact that you accumulate for constant sums ($s=x+y\implies y=s-x$).
The result is a smoother and more symmetric function. Smoother because you are averaging with positive-only coefficients, and more symmetric because of the mirroring (high functions values get a lower weight, and conversely).
The limit pdf is a Gaussian because this function enjoys a special property: the result of a Gaussian convolved with a Gaussian is another Gaussian.