It’s pretty fundamental in a variety of multivariate statistical methods. If the rows of X are multi variate observations, then XX’ is the Gram matrix (of dot products). This can be used in clustering and regression. If the columns of X are (centered) multivariate observations, then XX’ is a scalar multiple of the sample covariance matrix. This is used in PCA.

But in large scale applications you may not want to store XX’ but instead are interested in computing products of the form XX’ v on the fly.