To play a game on a PSP memory stick, it is only necessary to download the game onto the memory stick and navigate to that game in the PSP menu. This can be done with "homebrew" games or games purchased from the Playstation Network Store.
Download the game onto the memory stick
For a homebrew game, the PSP needs to first be attached to your computer with a USB cable. Download the game you would like onto your computer, then copy and paste that game into the "Memory Stick" folder for your PSP. For a Playstation Network game, navigate to the "PSN" store icon and press "X" to open it. Click "Games" and find the game you would like to play. Click "Add to Cart" and then click "Checkout." Enter your payment information and download the game.
Return to the main menu
For a homebrew game, disconnect the PSP from the computer by pressing "O" and then remove the USB cable. For a Playstation Network game, press "O" twice, which will return you to the main menu.
Navigate to the game and open it
For a homebrew game, find the Memory Stick folder on the PSP's main menu. Click the game that was saved from the computer and press "X" to play it. For a Playstation Network game, go to the games section of the PSP's main menu. Click on the game that was just purchased and press "X" to play it.