Basically for the English-language patches, some saintly person translated the Japanese version and "hacked" (for lack of a better word) the games so that the text he translated would be in English in the game.
Flash carts are wonderful little devices. Basically it's a device shaped like a normal DS cart that has a slot in it for a MicroSD card. On that MicroSD card you put the ROM file of the game (basically the game data extracted from the DS cart), and there's software on the flash cart that will play the game just like it's a regular DS game, and most flash carts have additional features like save states, built in Action Replays, and other things.
I hope this helped, let me know if you have more questions.