Teams -- in every sport -- typically add older veteran players with the idea that they'll be able to help their club in the postseason. If your team will likely make the playoffs, it's worth the gamble. In pro basketball, it's big men who can have the biggest impact while having the lowest risk.
Older players who are smaller in stature tend to struggle more than bigs after the 82-game grind. But you can get away with playing bigs as they can do a lot of things that don't show up in the box score: Set a screen, alter a shot, block out on a free throw, make the right read on a pass. And, of course, by nature of their size, they can still get 3-5 boards/game.
If the vet is PJ Brown or Robert Horry, they're still effective shooters, as well.
Yes, there are exceptions to the "big vs little" argument. Derek Fisher has been a great addition to the Lakers. On the other hand, Chris Webber didn't have much of an impact with the Warriors.