[DAMP] indicates early season dampening is in effect for fatigue calculations. What this means is that the effects of a higher than normal usage in the beginning of the season is reduced a bit as the actual usage balances out as the season goes on.
After your first couple of games many players may show fatigue even if they were everyday players in real life. The reason is that often times at the beginning of the season, players will rack up many plate appearances if your team scores a lot of runs or plays extra inning games. This will quickly balance out as the season progresses.