Thanks to zmcnulty from PSPvault.
Step 0: Follow the PSP to PC Basic Connection Guide
Before you start, you'll need to connect your PSP to your computer. A guide for doing this
is posted here.
Step 1: Navigate to SAVEDATA folder 
In your PSP's "PSP" folder is a directory called "SAVEDATA." Inside the SAVEDATA folder you will find something similar to the above. I am yet to completely figure out the naming scheme that these folders follow, but the codes do match the codes printed on the spines of PSP game cases. Anyway, and I think this goes without saying, but
different games use different folder names. In my screenshot above, ULJS00002 is a save file of the game "Kotoba no Puzzle: Moji Pittan."
And here's where things get interesting. You're probably asking "wait, so can I have more than one save game folder for the same game?" The answer to that question is
maybe. Implementing "multi-savegame" functionality is the responsibility of the developer. In Moji Pittan, for example, there can be only one save game stored on the Memory Stick at the same time. While many games implement profile systems to differentiate between different "save games" (note that it is in quotes because the physical files remain the same), many games do not. Certain PSP games have support for more than one save file, certain PSP games do not. Moji Pittan, for example, will only search for save games in the "ULJS00002" folder. Any other folder (even if it contains Moji Pittan save data) cannot be read.
Now that you understand a bit about the SAVEDATA folder's architecture, let's move on to transferring that save file you've just downloaded.
Step 2: Extract save game 
This process will vary depending on which software you use to extract .zip files, but I believe it's pretty straightforward. WinRAR offers the "extract here" option, which will simply extra the contents of a .zip file to the current directory.
All save game archives on PSP-Vault are only one level deep. This means that you needn't create any folders so the PSP can recognize your save files correctly, nor do you need to worry about moving save files to higher levels in your folder structure to transfer them to your PSP. On to actually transferring the save files, then.
Step 3: Transfer save game to folder 
As we mentioned, no folder or file locations need to be modified to use PSP save files posted on PSP-Vault. Just copy the folder you just extracted, and paste it into the PSP's "SAVEDATA" folder.
However, PAY VERY CLOSE ATTENTION to any folder overwrite warnings! Because the PSP save game folders use the same names for the same games, copying folders into these locations
could lead to lost data. For this reason, we recommend that you
backup your old save games before overwriting them. In my case, I would simply rename a folder like "ULJS0002" to "ULJS0002_backup."
Step 4: Exit USB Mode
Since your PSP has been in USB Mode throughout this entire process, just push X to exit.
Step 5: Verify transfer of save data
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To verify the integrity of the data you have just transferred, use the PSP's built-in Save Data Utility. Select it from the "Game" menu.