The calculation is the power method, but the main gist is modelling internet browsing behaviour as a Markov chain, wandering through randomly clicking links. The equilibrium probabilities of being at any page is the vector "r". I'm a bit surprised how few sources offer this explanation upfront, I guess they want to maintain the mystique, and want to make the technique appear more profound that it was.
Yup, I am very familiar with PageRank and the power method. Here’s my explanation from a decade ago: