The Federalists believed in a strong, centralized government. Having just emerged from the Revolutionary War, the new nation needed to take its place among the international community. To do this, it needed to pay off the enormous debts it had accrued during the war and develop a unified foreign policy. However, the United States was unable to pursue either of these aims under the Articles of Confederation, which had been deliberately designed to keep the central government weak. The framers of the Articles didn't want what a repeat of what they saw as the tyranny of British rule, so they devised a system that kept ultimate sovereignty in the hands of the individual states rather than a centralized federal government.
But not long after the war's successful conclusion, it became blindingly obvious that the Articles were no longer fit for their purpose and that a new constitutional arrangement would have to be made. As well as the issues of national debt and foreign policy mentioned earlier, domestic law and order was also a pressing concern for the delegates arriving at the Constitutional Convention in Philadelphia. If riots, uprisings, or any other form of public disturbance broke out in one state, there was absolutely nothing the weak federal government could do to stop the riots from spreading to another state. They would have to rely on the individual state authorities to restore order. But all too often they didn't, which is what happened during the notorious Shay's Rebellion. It was felt that only a strong federal government would be able to maintain good order in the United States as a whole.
Anti-Federalists were deeply suspicious of anything that smacked of centralized government. As passionate believers in republican democracy, they argued that ultimate sovereignty resided with the states and that states' rights should always come first. Anti-Federalists such as Jefferson believed that the precious liberties for which Americans had fought and died against the British would be jeopardized under their opponents' proposals. Liberty could only be maintained in a radically decentralized system of government. This way, it would be impossible for government to get too big and impose itself tyrannically on the states.
The final document that emerged from the 1787 Convention was, to some extent, a compromise between the Federalist and Anti-Federalist positions. Under the new Constitution, government became stronger and more centralized, but the incorporation of the Bill of Rights ensured that the states would still retain a considerable degree of autonomy in the new system.