In the UK, the main ones are run by The Restart Project (most of the London ones), and Repair Cafe (most of the UK ones outside London). Both these are international organisations - The Restart Project started in London, and Repair Cafe started in Holland. They work closely together in the UK. I do events for both of them (I was doing one this morning).
The other place you can find them are local maker spaces or hack spaces (some work via one of the two organisations above, some are independant). Some of these spaces have a regular open evening for non-members as an alternative to specific repair events.
Most of these work by voluntary contributions to pay for the venue and insurance. Small spare parts which the repairers carry are usually free. For more specialist parts, the owner goes and sources them themselves once they've been advised what's required. We have sometimes done specialist mobile phone/tablet repairs in conjunction with a commercial phone repairer, and then they bring stocks of replacement screens and the like so people can get those fixed in one go, rather than going off to get a replacement and coming back next time, but they charge for the screens - you won't get free screen repairs anywhere (unless its a warranty fault).