the 2600 was not the first console that came out lol
Quite complicated to explain, as wikipedia says "Homebrew is a term frequently applied to
video games or other software produced by consumers to target
proprietary hardware platforms (usually with
hardware restrictions) not typically user-programmable or that use proprietary storage methods. This can include games developed with official development kits, such as
Net Yaroze,
Linux for PlayStation 2 or
Microsoft XNA"
The first console wasn't the Atari 2600, the first real videogame console to ever been released was the Magnabox Odyssey, which was basically a pong machine, revolutionary for the time though, either way, going back to homebrew, remember that to create a homebrew you must to know all of the characters of a console you're intending to develop for, this includes knowledge about the Hardware, the Software, the limits and such, once you have all of the information, you can start programming things in new ways never done before.
To run homebrew, you need to mod either the thing you're using to play (Like a CD, or a cartridge, to put an example) or the product itself (Mostly consoles) in order to create homebrew, the history of Homebrew predates back in the 80s and 90s with the release of the NES, specifically, the NES versions of Brazil, Russia, China, etc. which included bootlegs of real games, as well modifications of them, those were the earliest examples of any homebrew probably made, and it's still on debate if that can be classified as such.
The rise of the homebrew began when the PSP was hacked, since Smartphones weren't as popular as today, people bought gaming consoles mostly to update their technology, and with the PSP a new world was discovered, it isn't called the Holy Grail of the Hacking consoles for nothing, multiple emulators, tons and tons of applications, and obviously, CFW's and backup loaders everywhere, this caused a revolution that it also affected the DS with the flashcards R4, which are among the most popular things ever in the world of hacking.
but the first to get homebrew AFAIK
It didn't, the Atari 2600 had Homebrew but not until reaaaaally later, the first console that had "true" homebrew, in some sort of way, was the Famicom/Nes, most respectively, their bootlegs version, which also included modified things from original games, this is probably the earliest example of any homebrew, if it can be considered as such.