How did Jeffersonian Republicans maintain their strict interpretation of the Constitution?

I think that watching Adams' administration as well as Hamilton's machinations, Jefferson and his fellow Republicans felt that his ascendancy to President represented a critical time for their philosophy and the country's future.  Armed with the classically liberal philosophy of laissez- faire, the Jeffersonian Republicans were able to point to the need to allow a decentered government and strict interpretation of the Constitution to hold.  The Adams' administration passage of the Alien and Sedition Acts were argued by Republicans like Jefferson to represent a dangerous interpretation of the Constitution.  Given Adams' negative view with the public, the idea of a strict interpretation of the Constitution gained some traction with the public.  At the same time, the perceived Hamiltonian collusion between banks and the government helped to strengthen the Republican claim of a decentered government that initiates more localized activity as opposed to a movement towards a nationalized banking structure in the government. The Republican claim of the Constitution favoring a more decentered view and one that was not so nationally centered was something that gained ground with the public, allowing the Republican philosophy of strict construction of the Constitution to become a part of the national dialogue and the Republican form of governance.

How did the Jeffersonians (Republicans) interpret the Constitution?

The Jeffersonians interpreted the Constitution very strictly.  They only wanted the federal government to have the powers that were explicitly given to it in the Constitution.

The Jeffersonians were suspicious of the power of the federal government.  They felt the federal government was more likely to be tyrannical and therefore wanted more power to be reserved for the states.  For this reason, they wanted to interpret the Constitution very strictly.  They wanted Congress to be able to exercise only its enumerated powers.  They did not want it to have any powers (like the power to create a national bank) that were only implied by the Constitution.