Oxygen in and of itself is a staling agent, plain and simple. Some styles benefit from being a bit stale. For instance, here in the states, what we know as "British" beer is typically a bit stale simply because by the time it gets here long after being brewed and having crossed the sea in a hot ship, it's not exactly fresh. So if you want to clone your favorite, say, ESB, then a bit of staling will help get you there.
Now, barrels don't exactly let oxygen in the same as if you left the beer exposed to air, or if you didn't fill the barrel completely. What they do is allow the beer (or wine) to micro-oxidize. Search the web for info about micro-oxidation and you can read lots of fun debate from the winemaking world about how it's either beneficial, bad, non-existent, or maybe no different than regular oxidation.