Well, they had Hamilton and Jefferson backwards, but they were quite right about the principle they were promoting. Federalism promotes greater state/region autonomy
in cooperation with a single national government.
Europeans use the word to describe a governmental philosophy of a stronger national government and weaker local governments, which is where some of the confusion may come from.
The Federalist Papers were written by Hamilton, Madison, and John Jay, which explained the political philosophy of a
need for a strong national government balanced by great state/local autonomy. Anti-Federalists opposed any kind of strong national government in favor of... well, anarchy and weakness (Articles of Confederation). They allowed the Constitution to be passed only with the addition of a Bill of Rights.
The Anti-Federalists were right about alot of things, though. Namely, decentralizing the national bank and an anti-French foreign policy...