Effectively, you flash an image on the SD which makes it multiple partitioned - a regular FAT32 one that can be read normally by the Wii (containing the loader and kernel, that are started via BootMii) and an ext3 or something with the main filesystem
The most widespread image contains Debian Lenny (5), I vaguely remember you can upgrade it to the next version (Squeeze) but not further (since the kernel would require manual compiling and updating, probably requiring in turn manual porting of the Wii drivers)
It's something fun to try (and you can run text-mode software reasonably, even more so considering old software is less bloated), but there is zero practical reason to use it (and I don't use "zero" lightly):
Around 2011 I used it as a 24/7 P2P machine (my desktop PC being from 2000, and without variable clock/voltages, up to 120 watts add up), but today a Raspberry Pi Zero W has over 4x the memory, yet it can be bought new at half the price of an used Wii, while using much less power, and pretty improved wifi/bluetooth/expandibility/software support!
If you agree with my theory of the homebrew golden age (= the amazing selection and variety of amateur software of DS/PSP/Wii was caused by the fact they were the only cheap and widely available pocket devices with wifi/point-and-click device working with any TV) it's not surprising any console Linux port dies far faster than homebrew in general for the same console
them