I heard there is an archive by an organisation or something or other.

Sometimes the maker of the game offers a ROM for download, either for purchase or for free. There are USB devices that'll dump a ROM cartridge, such as the Retrode.

Sometimes it's not an "emulator" in the typical sense. Some of these are hardware reproductions (see: Palmer Luckey's ModRetro Chromatic).

Others do hardware emulation using FPGAs (see Analogue Pocket).

Even a lot of the emulator-based hardware include a physical cartridge slot. They read the data off the cartridge and then run it on an emulator.

Presumably they just download the original ROMs which are incredibly easy to get just by searching "gameboy ROMs". Because they are so incredibly small in filesize they are hosted pretty much everywhere. People share still photos with each other that are larger than most of those games.

iirc most of these type of things come preloaded with all of the games up to GBA or so, the entire library is tiny by today's standards, I download ML models bigger than like... all games made prior to 2000 on a regular basis

1. torrenting sites

2. archival sites