What were two big reasons as to why Anti-federalists opposed the idea of the Constitution? What was promised to Anti-federalists to eventually sway enough to support the Constitution?

The Anti-Federalists opposed the Constitution because they thought it vested way too much power in the hands of the federal government. The American colonists had just fought a war against what they saw as British tyranny. The last thing they wanted to see was one kind of tyranny replaced by another. The American Revolution had been fought on the principles of radical republican liberty, which meant, among other things, limited government and the primacy of states' rights. Both of these principles were thought by the Anti-Federalists to be undermined and threatened by the Constitution.
Anti-Federalist hostility to the Constitution was also based on economic grounds. The Anti-Federalists tended to represent agrarian interests and believed that the framers of the Constitution were too close to the banking and commercial sectors of the economy. They were concerned that the federal regulation of commerce would mean central government overriding the concerns of rural folk, who were, after all, the vast majority of the country at that time. Anti-Federalists looked upon their opponents as unashamed elitists concerned only with protecting the interests of a wealthy minority.
The very real concerns of the Anti-Federalists led to the passing of the Bill of Rights; in essence, this was a compromise on the part of the Federalists to try and reassure opponents of the Constitution. Anti-Federalists felt that not enough attention had been paid in the Constitution to the inalienable rights for which countless Americans had recently fought and died. Rights were implied, not explicitly set out. So they demanded, and eventually got, a formal guarantee of those rights, clearly enshrined in the Constitution as the Bill of Rights.
https://teachingamericanhistory.org/resources/bor/fed-antifed-debate/