Well, the Framers did not all have the same intentions, in fact their opinions on the matter of government power differed sharply, so let's look at the eventual compromise they came up with.
The power of the King was divided between three separate branches, the Executive, the Legislative and the Judiciary. We can draw a few inferences from that fact alone, as it is obvious that the Framers in general did not trust centralized government, and while they realized they needed one with the authority to defend the country and pay the bills, they also feared the power of a King, and wanted to create institutions that would dismantle that power.
By creating checks and balances, we can infer that the Framers feared power would be concentrated again at some point unless they included safeguards against it.
The Framers developed the system of checks and balances so that the three branches of government (legislative, judicial, and executive) could check each other. For example, the Congress can override Presidential vetoes, and the Senate has the power to approve certain Presidential appointments (and there are many other ways the legislative branch can check the powers of the other two branches). The President can veto bills passed by Congress, and the judicial branch can declare laws passed by the other two branches unconstitutional. These are some of the ways in which the branches of our government can check each other.
The division of government into three branches was first developed by the Enlightenment philosopher Montesquieu. His idea was to limit the power of monarchs. It can be inferred that the Framers of the Constitution also feared the concentration of power in the hands of one leader or one branch of government, as that leader or group could become tyrannical. The Framers protected against what they believed was the tendency of leaders to become tyrannical by building checks and balances into the system of government.