I watched the final game of the World Series yesterday, the twelfth Fall Classic that I've seen. This year, I scored the World Series as well.
The first time I filled in a baseball scorecard was at a Phillies game in the now demolished Veterans Stadium in Philadelphia in 2002. I think we got them going through the turnstile and I made an untutored stab at completing it.
There are two reasons to score a baseball game. The main one is to keep track of at-bats, what batters did in their previous trips to the plate, who's on-deck and the pitcher's record of walks, strikeouts etc. But it's also fun. As well as being creative, using your own symbols and adding a "!" for an outstanding play, I also find jotting down the K's, L5's and 6-4-3's adds to the rhythm of watching a ballgame with its three outs in the top and bottom half of each inning.