The TL;DR version probably isn’t going to make any sense unless you’re a computer programmer, but in brief:
The Nintendo 64 has a plug-in accessory called the Expansion Pak that adds 4MB of extra RAM to the system. Most N64 games don’t use it; a small number require it.
Paper Mario suffers from several bugs related to menu loading that, in combination, can cause the execution pointer to go shooting off to gods-know-where. One of the places the it can go shooting off to is a normally nonexistent memory address that happens to correspond to a particular block of RAM on the Expansion Pak, if it’s installed.
By itself, this isn’t helpful. Jumping the execution pointer to weird places often allows for arbitrary code execution, but Paper Mario is one of the games that doesn’t use the Expansion Pak, so there’s no way to write data to it during normal play.
However, because it doesn’t use the Expansion Pak, it also doesn’t initialise the Expansion Pak’s memory on boot. This is important because the Expansion Pak consists of a relatively high-persistence type of RAM that can retain its contents for several seconds without power.
If you know what an execution pointer is, you’ve probably already figured out where this is going: we need to deposit a useful payload on the Expansion Pak using a different game, then quickly swap cartridges to Paper Mario and do the menu glitch that jumps the execution pointer to the Expansion Pak’s address.
It turns out that the quickest game to set this up with is Ocarina of Time. By doing some very stupid tricks involving stale reference manipulation of a linked list structure containing data about fairy sparkles, you can drop a few bytes of arbitrary code on the Expansion Pak. It’s not much, but it’s enough for the next step.
Swap cartridges to Paper Mario and do the menu thing. When that crafted payload is carried out, it causes the execution pointer to jump again, this time to the memory address containing the name of your save file.
The name of the save file in turn contains a series of nonsense characters corresponding to instructions to update your story progress flags to right before the end credits cutscene, autosave, then crash the console.
Restart console, load doctored save file, watch credits.
Or, to TL;DR the TL;DR: you use a glitch in Ocarina of Time to deposit a logic bomb made of fairy dust on the N64 Expansion Pak, then boot up Paper Mario and do stupid tricks with the menus to ricochet the execution pointer off that payload and start executing your save file’s name as code, thereby enabling arbitrary code execution.