Massively complicated, but at its heart quite a basic idea. As with most of the things we look at, the most important thing to do is to ask the right question.
Prerequisites for understanding: Regression to the mean.
Prerequisites for derivation: Environmental effects (park and league), data.
We might know what a player did today, but how do we know what he'll do tomorrow? Even after applying vigourous corrections to statistics (we use regression and environmental adjustments in particular) and coming up with our best estimates of talent level, we're still left exposed to actual changes in talent. It's not like this is an uncommon occurrence either: 'True talent' changes all the time as players get better and worse, and the primary reason for this is age-related development or decline. For our purposes, we'll treat increased likelihood of injury as part of age related decline, mainly because it is. Trying to isolate injury from pure aging will lead to significantly different results, and it's pretty pointless because we then have to introduce an injury adjustment as well.
So we accept that we need to understand the effects of aging on performance on more or less every baseball skill in order to project future performance. We need to know how age impacts a batter's eye, a pitcher's fastball, even a fielder's range. The question, obviously, is how we actually do it.
Let's stick with what we know. We are (or should be) familiar with the method of deriving MLEs by comparing players between leagues as they are promoted or demoted until we can build a chain from the minor leagues up to the majors. Why not do the same thing again? Choose a set of players for each age group with a certain (large) sample of plate appearances/pitches/defensive opportunities/whatever, park and league* adjust their numbers, and see how they compare between adjacent years. Put everything together and voila! Aging curves, right? Well, for the most part. There are arguments to be made for massaging the sample in different ways in order to strip out selection biases, as well as for taking different approaches than a simple year-by-year comparison. However, you're going to come to a pretty similar conclusion no matter what you do, so having knowledge of the basics is probably fine.
- Hitting follows a nice steady curve, peaking in the late twenties and then slowly tailing off
- Pitching does not. Instead of a steady rise and fall, pitchers seem to reach plateaus of ability, then jump/fall to a new plateau.
- Fielding peaks earliest of all the major skill groups
- Different skills in groupings do not decline at the same rate. Batting eye improves for a long time, but contact ability does not.
- Certain skills falling off fast means that we should be paying close attention to players who are marginal in those skills. A high-contact guy losing some hitting ability does not kill his value, but an awful contact hitter who gets worse may stop being a viable major leaguer.
*Including adjusting for changes in specific league quality year to year - the game's always improving.