There really isn't a 'game' going onto the memory card, moreso just lines of text and a program that loads them. What's going to happen is InternetExplorer (Dan Salvato) is going to write an app for the homebrew channel that writes to your memory card a special save file for Melee. When you play Melee with this special save file, the game is tricked to load IE's program that loads Gecko codes (Gecko codes are similar to Action replay codes - They change memory addresses in the game. You can unlock and change things this way). One the Gecko codes are loaded the game is rebooted with them and the game is revolutionized.
So no, you won't directly be putting a file onto your memory card (I'm sure it's possible with something like a WiiKey Fusion, but I'm not going to get into that). Unless you purchase a premade card from IE, you'll need to softmod your Wii, install the homebrew channel, download and install IE's app and then inject the modified save file onto your memory card.
Buuuuuuut it get's better. Since the modified save file is technically still a save file, (it's a .GCI if I recall correctly) the save can be copied to other memory cards natively though a GameCube or Wii's memory card management system (So basically if your friend has a memory card with the hacked save file, and you have to do is practically copy and paste it on the console - your console for copying it doesn't have to be modified AT ALL. As you can probably guess, since the end product is just a hacked save file it can be loaded on any modified or unmodified system.)
Hopefully that explanation helps.