After the war, the former colonies decided that sticking together militarily and speaking with one voice on foreign policy had worked out pretty well for them -- they'd fought a war against the world's number one superpower and won -- so they ought to continue to do so, and the new government ought to set up uniform rules for the way states trade with each other, and establish a single place to register scientific inventions, etc. Sort of like the European Union today, or maybe the European Community that preceded it.
The states had just thrown off the shackles of one overlord and weren't about to put themselves under another. After these concerns resulted in an initial agreement so weak as to border on totally ineffective [1], their diplomats tried again [2] [3], and the states agreed but quickly demanded more limitations on federal power [4].
The ideological divide between states that allowed slavery and states that did not was a crisis on slow boil for the better part of the next century, resulting in various political compromises [5] [6] but eventually leading to a major civil war [7].
After the war was won by the side that favored a stronger national government, one of the side effects of the event was setting a precedent of the federal government imposing its policies on unwilling states at gunpoint [8]. (It's not a typo in that reference when it notes the Republicans pushed for radical expansions of African-American rights; both parties basically completely switched sides with respect to racial issues after World War 2 [9]).
The sense of national unity was strengthened through the outcome of the war, increased interstate commerce due to the inventions of the Industrial Revolution, and the crises of the Great Depression and the world wars. All of these events gave the federal government reasons to expand its powers [10] [11] [12], in ways which seemed like good ideas at the time but ended up resulting in a much more powerful federal government, and much weaker state governments, than anyone had envisioned.
[1] http://en.wikipedia.org/wiki/Articles_of_confederation
[2] http://en.wikipedia.org/wiki/Philadelphia_Convention
[3] http://en.wikipedia.org/wiki/United_States_Constitution
[4] http://en.wikipedia.org/wiki/United_States_Bill_of_Rights
[5] http://en.wikipedia.org/wiki/Three_fifths_compromise
[6] http://en.wikipedia.org/wiki/Missouri_compromise
[7] http://en.wikipedia.org/wiki/American_civil_war
[8] http://en.wikipedia.org/wiki/American_civil_war#Reconstructi...
[9] http://en.wikipedia.org/wiki/Southern_strategy
[10] http://en.wikipedia.org/wiki/Incorporation_of_the_Bill_of_Ri...
[11] http://en.wikipedia.org/wiki/Sixteenth_Amendment_to_the_Unit...