In general, free oxygen will bind to various other compounds in your beer and create off-flavors -- not sure I've ever had it happen to my beers, but it's commonly described as wet cardboard, or less frequently, sherry, rotting garbage, and many other things that your beer is not supposed to taste like. This is called oxidation.
Very early in fermentation, in the "lag phase" (before you start getting visible krausen/airlock activity) the yeast will quickly absorb any free oxygen -- they need it in order to reproduce up to the full number of cells you want fermenting your beer. This has the obvious benefit of giving you enough yeast for a good fermentation, but it has the happy side-effect of pulling all that oxygen out of the wort before it has a chance to oxidize anything else.
After the yeast are done reproducing and are just gobbling up sugars or have finished up the fermentation and dropped out of suspension, they don't need the oxygen, so, it stays in the beer and oxidizes it.