Hamilton was distressed by the crippling debts held by the states and the federal government's limited ability to finance itself. Madison, and southern agrarian states, opposed Hamilton's plan to assume the debts of the states and strengthen federal taxing powers.
Southern states feared a strong federal authority, dominated by northern financial concerns — many of whom speculated in purchasing Revolutionary War debt for pennies on the dollar. Hamilton feared the collapse of a young Republic lacking financial credibility, unable to finance regular operations of government, including providing for its defense.
The location of the capitol was more than a question of geography — it served as a symbolic matter of trust between the mercantile North and the agrarian South; a northern economy increasingly reliant on manufacturing and commerce, and a southern economy still reliant on the fruits of the land.
The dinner resulted in a grand bargain that would locate the nation's capitol on the shores of the Potomac River — in the South. And that agreement (combined with a sizable tax break for Virginia) allowed Madison to free up votes in Congress for Hamilton's assumption plan.
The dinner table bargain was one of the critical moments in American history. It didn't solve every problem — it would take a bloody civil war 60 years later to cement the Union. But it helped to bind the states together, and to a federal authority, at a time when it could have easily fallen apart.