The reason for this is that many of the provisions of the Constitution are much more conservative and less radical than the principles of the Revolution, as seen in documents like the Declaration of Independence.
Because the leaders of the Revolution needed mass support for their movement, they emphasized such things as democracy and the equality of all people. These ideas were largely followed by the governments under the Articles of Confederation. But this system did not seem (to many of the elite) to be working and so they pushed for a new constitution.
The Constitution of 1787 was much less democratic than governments had been under the Articles. The Constitution, for example, only allows the people to vote directly for one part of the federal government. Under the Constitution (before amendments) the people only voted for the House of Representatives while the President, the Senate, and the Supreme Court were selected without direct public input. There were also provisions of the Constitution that limited what state governments could do, like creating laws that would forgive debts (Contract Clause).
In short, the Constitution was meant to some extent to take power away from the people so there would not be as much democracy. It was not trying to create a monarchy or a dictatorship, but it wanted more room between the people and government so the government could do what it knew was best rather than following the whims of the people. This represented a move away from the democratic ideals of the Revolution.
It is for this reason that Henry said the Constitution was a counter-revolution.