Where does the fan keeping score tradition in baseball come from?

One could say this is based on tradition, but there is more to it than this. Most people who keep track of the score, write down a lot more information about the game; and not just the number of the runs scored. This way they can have a detailed data about things such as types of pitches and stolen bases during the game. Otherwise you would need to wait for the game to finish and then check for the stats.

It also helps you to focus more on the game, and you will also be able to look at the data for a game that you have attended any time you want (having a database of your own), or checking for players stats.