As seems to happen so often in politics, the Federalists opposed the Louisiana Purchase not on principled grounds but because they thought it would hurt them politically. They claimed to oppose the Purchase because it was unconstitutional. But they really opposed it because they thought that the states that would be made out of the Purchase would be more Democratic-Republican than Federalist.
In general, the Federalists were an elite party of New England merchants. They had some support in other states, but almost solely among the upper classes. The Federalists reasoned that the states that were made out of the Purchase would be settled by small farmers, a class that was solidly in favor of Jefferson's party.
The Federalists claimed they opposed the Purchase on constitutional grounds. They said the Constitution did not give the president the power to do things like buying territory. This was somewhat hard to believe, though, since the Federalists were the party that was in favor of interpreting the Constitution broadly. Because their complaints about the Purchase were so contrary to their usual beliefs, historians believe that their real motive was the fear of losing political power.