I first encountered this technique on a James Hoffmann video and then through the power of THE ALGORITHM I saw a few more videos pop up. The idea is that the paper filter forces the water to travel through the puck more evenly and causes a more even extraction, while simultaneously filtering out some oils and fine bits of coffee.
James Hoffman's "How I Make Espresso: Tools and Techniques"
https://www.youtube.com/watch?v=xb3IxAr4RCo
It seems like everybody uses a particular 2.25" hole punch from Amazon to punch circular filters out of a chemex filter paper. Becuase I use a Breville Barista Express with it's ridiculous non-standard 54mm portafilter, I can't do that. I've seen reports that a 2" filter paper will get the job done but I haven't tried that myself. Instead, I took an aeropress filter and put my double filter basket on top of it and cut it out with a hobby knife. Super time consuming and thus not viable for day to day use, but it's good enough for science!
https://www.amazon.com/gp/product/B00DJJR3RW
It's pretty cool. Using Monastery Coffee's San Jerónimo Miramar, the filter seems to completely cut the typical espresso bitterness leaving an extremely fruity, acidic cup. I'd say in this particular coffee, the level of acidity may be too high for my liking but I'm now keen to try a more caramelly/chocolatey single origin to compare!