It's described at a high level in media often. Here's more detail though:

Person buys a product, or downloads some software - that unknowing to them, was reaching out to a central server elsewhere to establish a network tunnel. And over that tunnel, the other server routes traffic onto that person's home network. Which can then of course go onto the internet, and it appears like the traffic originated from their home internet connection.

Other people rent access to this network. The goal being, you have surreptitious access to appear as any one of tens of thousands of home network IP addresses on the public internet. Bad guys can use this to rotate through many different IP addresses when their access gets blocked because of malicious (or benign but unwanted) behavior on the internet. Residential IP addresses are more 'valuable' because they don't have the negative connotation that datacentre and commercial IPs can have, and they can be spread geographically or across many ISPs.

A very common way for people being a part of these residential proxy networks is by buying dodgy streaming boxes, and also installing shady apps on TV app marketplaces.

This isn't a theoretical problem either. My employer's website was being hammered by residential proxies a month ago. It wasn't until we setup WAF captcha rules that they mostly subsided. It wasn't even the typical exploit scanner. I don't know what the purpose was but it was effectively a DDoS.

I was expecting that VPN’s (maybe not all) also use their own customers as relays, but you don’t mention them. Am I wrong?