What differentiates a simple democracy from a republic is the complex system of checks and balances that the latter employs to promote both liberty and stability. In the federal American Republic, authorities are divided vertically between the states and the national government. Powers are also separated among the President, Congress. and federal Judiciary. The Constitution is in essence a charter for dividing decision-making power.
A decision-making charter requires dispassionate enforcement. The political issues that stir people’s souls are almost always substantive rather than decisional. People march for and against abortion rights, not in defense of the appropriate constitutional entity to make that decision. But the locus of the decision is one of paramount importance to the maintenance of a republic, not least because of the danger that one branch of the federal government or the federal government as a whole will usurp power, creating more dangers of tyranny.
Thus, a central underlying issue for a republic is how to assure that those decisions about the proper authority are made in a neutral, dispassionate way—not swamped by the emotions generated by substantive disagreements. The first line of defense is to choose the most dispassionate institution to make these meta-decisions (i.e., the decisions about who decides). When Alexander Hamilton said that constitutional review should be lodged in the judiciary because the judiciary embodies judgment rather than will, he was emphasizing the comparative dispassion of the third branch.