The way I understand this is that adding of random variables is a smoothening operation on their densities (more generally the distributions, but let me speak of densities only).

A little more formally, additions over random variables are convolutions of their densities. Repeated additions are repeated convolutions.

A single convolution can be understood as a matrix multiplication by a specific symmetric matrix. Repeated convolutions are therefore repeated matrix multiplications.

Anyone familiar with linear algebra will know that repeated matrix multiplication by a non degenerate matrix reveals it's eigenvectors.

The Gaussian distribution is such an eigenvector. Just like an eigenvector, it is also a fixed point -- multiplying again by the same matrix wil lead to the same vector, just scaled. The Gaussian distribution convolved is again a Gaussian distribution.

The addition operation in averaging is a matrix multiplication in the distribution space and the division by the the 'total' in the averaging takes care of the scaling.

Linear algebra is amazing.

Pagerank is an eigenvector of the normalised web adjacency matrix. Gaussian distribution is the eigenvector of the averaging matrix. Essentially the same idea.