The polynomial Freiman-Ruzsa conjecture is one of the important conjectures in additive combinatorics. It asserts than one can switch between combinatorial and algebraic notions of approximate subgroups with only a polynomial loss in the underlying parameters. This conjecture has also already found several applications in theoretical computer science. Recently, Tom Sanders proved a weaker version of the conjecture, with a quasi-polynomial loss in parameters. The aim of this note is to make his proof accessible to the theoretical computer science community, and in particular to people who are less familiar with additive combinatorics.