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There is a player statistic called Win Share that attempts to estimate how many wins a specific player contributed to the team. There are various flavors to the statistic but by one of them last year Manny Ramirez contributed a total of 11.3 wins to the Red Sox/Dodgers last season.
In about half a season, he contributed 6.3.
If you replace him with an average outfielder you'd expect that average person to have contributed about 2 wins (4 over the season). So, if you assume similar production the Dodgers might be expecting Manny to provide an extra 7 wins or so in a full season (assuming an average replacement was available).
In baseball, even though they play a lot of games, 7 is a HUGE number. Last year, only 2 of the 6 playoff teams would still have made the post-season if you subtracted 7 games from their win column (Angels would still have won division by 14 games and the White Sox might have become a wild card team). Of the 24 teams that did not make the playoffs, 8 would have if everybody else stayed the same and you gave them an extra 7 wins.
Whether that is worth $25 million I can't say but apparently it is to the Dodgers. Keep in mind, though, that while baseball is a for profit business that doesn't mean that the irrational desire to win at the expense of pure profit is completely removed.
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