The gist is that you can formulate the PageRank problem as a flow equation r = Mr, where r is a vector of PageRanks and M is a stochastic adjacency matrix, which can be interpreted as the probability of visiting one page from another.
This matrix is huge and hence this is tough to solve.
But since M is a stochastic matrix, it has the property that its largest eigenvalue is exactly 1, and all other eigenvalues are < 1. And the equation r = Mr can be interpreted as r being the eigenvector of M with eigenvalue 1, as this equation implies multiplying by M is equivalent to multiplying by 1.
Repeatedly multiplying by M hence shrinks every eigenvector except the eigenvector with eigenvalue 1. So you start with some initialization of r, and repeatedly multiply by M until it converges. The resulting value of r solves r = Mr since all other eigenvectors shrank towards zero.
This is a much more efficient way to compute r, and hence makes PageRank computationally feasible.