Why did the tension between the demands of national security and the protection of citizens' rights become so severe during the 1790s?
Because at that time, the first decade of the new republic, the country was deeply divided between Federalists and anti-federalists, and they didn't trust each other. They had agreed to the Constitution only because there were checks and balances on government power and a Bill of Rights that protected them as individuals. Even then they were still suspicious.
Washington made them more suspicious when he dealt with the Whiskey Rebellion in 1794. He sent 12,000 troops marching into western Pennsylvania to deal with an uprising that was largely over by the time they slogged through the mud. It sounded, to them, like something the King would have done.
Our ships were being seized left and right by both the British and the French in the 1790s, and they were easy targets since we had no navy to protect them. The Federalists, especially Alexander Hamilton, wanted one, while the anti-federalists said we would turn into England. Then when Adams passed the Alien and Sedition Acts (unnecessarily so, as the threat from immigrants was minor - mostly was a threat to his re-election) the rift between the two sides got even wider.
The reason for this was that the United States was a very new country at the time and was not yet used to the ways of democracy. Because of this, there came to be a feeling on the part of the government that opposition to the government was the same thing as opposition to the existence of the country. Today, we know that a person or a group can (for example) love America while hating the Democratic Party. In the 1790s, people were not yet used to this idea. They were also having to worry about it at the same time that a war between France and Britain was causing a rift between American supporters of each side. It was for these reasons that the Alien and Sedition Acts were passed, infringing on the rights of citizens in the name of national security.