(playing them directly from the site doesn't work, luckily LeoV in the Flashpoint Discord was able to get all three working correctly through hacking)
All three games feature SpongeBob characters and have to do with caps and collecting them. Based on how well you perform in each game, you get a cap of a certain level. In the end screen, there is a "Pick New Caps" button, but clicking it freezes the game. You can play the working games for yourself here.
Aside from a couple of files on the Wayback Machine, this blog post is the only other reference to Nickelodeon's Happy Caps I could find on the internet, and it contains an image of an "unused game screen" that consists of an entirely different game from the three on Wayback. Perhaps there were more games that can't be found through the Wayback Machine?
I have no idea what Happy Caps is or what its purpose was, but judging by the dates of the Wayback captures the event took place sometime in mid-late 2008. If anybody remembers these games or has info on Happy Caps, please let me know.