Question

Why did the Antifederalists oppose the new constitution?

Answer 1

Because it didn't include the Bill Of rights. After that was added ( the 10 amendments ) they were happy with it.

Answer 2

Generally they thought the States would be giving up too much power to the national government. They had just gotten away from George III, and didn't want too go through that again. Anti Federalists didn't want a President with too much power.
Anti-Federalists today are Tea Party folks, Libertarians and people who point out how the President is trying to get around Congress. We have always had the sentiment. John Adams was accused of being a Royalist. There was some grumbling about George Washington's use of power, but nobody was very public about it. One did not criticized George Washington. Sorta like simply walking in to Mordor. It was simply not done.
And yes the Bill of Rights was the compromise piece promised to the Anti-Federalists. But they weren't all balloons and smiley faces about it.