The important thing to know is that the passage of the Alien and Sedition Acts represented the "high water mark" of the Federalist Party. Put another way, it rose and fell with these two acts.
The Federalists had always controlled the Senate, and the 1794 Congressional elections gave them control of the House of Representatives. Finally, in 1796, John Adams beat Thomas Jefferson for the Presidency by three electoral votes, with Adams getting the five New England States, New York, New Jersey, Maryland and Delaware, and Jefferson getting six southern states plus Pennsylvania.
The Federalists represented the business and banking interests of the Northeast United States, in the manner of Alexander Hamilton, while Jefferson represented the southern, agrarian interests. The Federalists justified the passage of the Acts through Hamilton's doctrine of "implied powers" of the Constitution. Lastly, the Federalists had benefited from foreign developments; the French Revolution and a near-war with France in 1797 strengthened the hands of the pro-British, pro central government, Federalist against the pro-French, "states rights" Jeffersonians.
The "countervailing" coalition of southern agrarians and northern immigrants defeated Adams and the Federalists in 1800; New York state "switched sides" and gave its electoral votes to Jefferson (and Aaron Burr), meaning that Jefferson and Burr each had eight more electoral votes than Adams. After being named President by Congress, Jefferson let most of the Alien and Sedition Acts "sunset" in 1801, and pardoned most people (his allies) that had been fined or imprisoned under them.
As to why the Supreme Court didn't do anything, it was not a full "third branch" of government at the time of the Alien and Sedition Acts; in fact, not until the principle of judicial review was established by Marbury v. Madison in 1805. Recent Supreme Court decisions on free speech suggest that the Alien and Sedition Acts would have been found unconstitutional by a fully empowered Supreme Court.