The main thing that convinced many (but by no means all) of the Antifederalists to support the new Constitution was the addition of the Bill of Rights. This alleviated the fears of many of the more moderate Antifederalists.
Antifederalists were very wary of the power of the federal government. They had been relatively happy with the system that the Articles of Confederation had created. They felt that the federal government was much more likely to abuse the rights of the people than the state governments would have been. They worried that the federal government might, in the future, become like the English government. Because it was less closely connected to the people, it might start acting for the benefit of national elites and forget the rights of the people in the states.
The Federalists could not completely convince the Antifederalists that this would not happen. However, they did promise to amend the Constitution to include what we now call the Bill of Rights. The Bill of Rights would explicitly protect the freedoms of the people from the national government, thus making it less likely that the national government could abuse people in the way that the Antifederalists feared.
By adding the Bill of Rights, the Federalists managed to convince enough Antifederalists to support the new Constitution and the document was ratified in 1789.
A Bill of Rights was added in 1791. In part to gain the support of the Anti-Federalists, the Federalists promised to add a bill of rights if the Anti-Federalists would vote for the Constitution.