How do game designers come up with suggested (or minimum) age requirements for games?
According to this and this, the government has a role in labeling of products that can be used by children, and that 13 is the youngest recommended age you can put without doing additional testing (though sometimes even that is not sufficient). The lower the age, the more robust the testing required, particularly for things recommended under age 3. The rules were different when Squad Leader came out, which could be why it says 14+.

The general rule of thumb in the industry seems to be to put the highest age you can that you don't think will adversely impact your marketing. For Pandemic, this is pure speculation, but if it were me, I would assume that any parent who's OK buying a game for their family in which you can lose and the world's population will succumb to disease either knows their children well enough to know if they can handle it or doesn't really care about theme in games (either way, 13+ probably wouldn't scare them either). Gamewright's core demographic is parents of young children, and they're sold in more mainstream stores than Z-Man Games (generally speaking), so it would make sense that they'd want Rat-A-Tat-Cat to be listed as young as reasonably possible to actually play the game, even if they aren't getting the most out of it.