I only read the article halfway through, and I am no physicist, but my understanding was that this was a new mathematical method that allowed to calculate past the divergent part.

Possibly because some terms cancel out later? a bit like some limit calculations.

If that's the case, I was picturing it a bit like how imaginary numbers were initially introduced, to find real-valued solutions to 3rd+ degree polynomial equations. Step into another realm, perform your transformations, and find back a real solution. Laplace transforms come to mind as well, there are a phletora of such tools (fourrier, taylor series, etc) that allow to express the problem in a different space.