It is not about a political party. It is the old, demanding idea that a free state survives only if its citizens have the virtue to put the common good above themselves.
Republicanism, with a small r, is the political philosophy behind a republic: that a free government depends not just on good laws but on the character of its citizens. The word has nothing to do with any modern political party.
At its core is the idea of civic virtue. A republic asks citizens to be active, informed, and willing to sacrifice private interest for the public good. Where a monarchy relies on the king, a republic relies on the people, which means the people must be up to it.
It carries a constant warning about corruption. Classical republican thinkers, whom the founders studied closely, believed republics die from within, when luxury, selfishness, and apathy erode the virtue that holds them together. Vigilance is the price of survival.
The founders saw themselves in this tradition, looking back to Rome. They feared that Americans might lose the civic spirit that made self-government possible, which is why they wrote and worried so much about education, participation, and public character.