Most of the history is documented on the hackmii blog. It went something like this-
First other was the mod chip allowing you to play backups of retail games.
Then other was the twezzer attack where the common key was extracted from a running Wii with a pair of twezzers, with that they could decrypt the contents of Wii disc and examine the files.
Then the fakesign bug "trucha bug" was discovered, allowing custom code to be loaded on a Wii via disc with a modchip.
Then the twilight princess buffer over flow was developed allowing code to be loaded from an SD card without a modchip.
Several homebrew loaders where made, like loadmii, but the HBC channel became the king as an installable channel.
The trucha bug was killed with system menu 3.3, but it could be restored by downgrading back to 3.2. Many more iOS and system menu exploits where found. To hack the Wii you need two exploits, one to load an app, another to install to nand. There have been a lot of exploits, many have been fixed with system menu updates, just to be replaced by another.
Most of the debugging to do this is done with a USB gecko.
That is about all I know, the exploits are well documented at wiibrew and the hackmii blog has many interesting posts following the above developments.
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