Ultimately, the system of checks and balances was intended as a check against government overreach and abuse of power. Remember, the American Revolution was fought during an age where European politics was still dominated by monarchies, and the revolutionaries themselves framed the revolutionary struggle as a fight against tyranny. However, this raises the question: How would you prevent the government of the United States from itself descending into tyranny?
Interestingly, the Constitution was not the first system of government planned for the United States. The first, envisioned in the Articles of Confederation, actually sought to address this problem by making the federal government exceptionally weak (in comparison to the state governments), but this had the effect of making that government too weak to effectively address the challenges of the time. Thus, the framers understood that a stronger, more centralized governing structure was necessary, but at the same time, this greatly raised the potential for future overreach on the part of the federal government itself.
The framers addressed this problem by dividing power between the three branches of government (the judicial, executive, and legislative branches), with each serving as a counterbalance against the others. Additionally, it should be noted that this system of checks and balances has itself evolved throughout US history.