To me, one of the most fun and satisfying part of watching a baseball game at the ballpark is to keep score. You don't control the game at all (though I always imagine that I can predict how my half-pencil will move the next at-bat. Come on 1B...just one more hit...). I especially enjoy the fact that you get an instantaneous snapshot of how the player has done in previous at-bats and innings. It also gives a brief mental picture of each event of the game. Never before can three symbols mean so much (aside from other acronym types) like GO 5-3. Can't you just picture the event?
The cumulative effect of keeping score and the statistics that are present by the end of the game are the value. With one glance, you can see what events led to the ultimate goal, scoring runs. You can calculate BA, OBP, SLG and add totals to other things like HR and RBI to get a picture of how a player contributed to the team scoring (or not scoring) runs. Pitching statistics are also apparent: K, BB, H allowed, ERA, RA, etc. You can see with your eyes and then correlate the game with the numbers.
A season is an accumulation of games and a career is an accumulation of seasons. How to figure out how a player contributes to scoring runs, preventing runs, and ultimately wins (or wins in an average or neutral environment) gives a good idea of the player's value. There are many ways to measure this. Some of them are cumulative stats based completely off a player's total and rate stats (win shares, base runs). Others are based on comparing the player to a level of his peers (EqA, VORP from Baseball Prospectus, WARP3, RSAA). Others are based on comparing to peers adjusting for their environment to compare players in different eras (OPS+, ERA+, Neutral Wins). These all require some calculation. But they are based in the one reality that baseball exhibits, scoring runs is good on offense and preventing runs is good on defense.
Even basic statistics that keeping score and the boxscore show have value. They give a quicksnapshot of a player and how well he took advantage of opportunities. They also are the basis for any other comparative statistic that might exist. Build a base with actual events and correlate this to value and essentially to runs/preventing runs. It seems simple, but there are always roadblocks to devising a number that encapsulates as much as you would like to capture.
I don't have the time to invent new statistics or sift through vast amounts of data. There are many sites and books and organizations that do that very well. It is of interest to dive deeper than keeping score and reading the boxscore to appreciate baseball on a historical and reflective level. It is always interesting and always sparks debate, which many people enjoy. A worthy debate is almost as good as the reasons for it existing.