How does that work exactly? If federal laws are above state laws, then what's the point of state laws?
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...
When FDR became president, he took economic desperation as a blank check to do anything to fix the Great Depression. The federal government explicitly had no power over local manufacturing and employment, so they created the power by reinterpreting anything that hypothetically might affect later interstate commerce as if it were interstate commerce. So birthing too many piglets was the same thing as selling too much smoked pork across a state line, and the federal cops would come to your farm and correct it by killing the unlawful piglets.
This should probably have resulted in the rape of Washington, D.C., but at the time Americans were starving in Bangladesh-style tent cities and were willing to put up with a lot. Also local governments were flying the Communist flag and high federal officials were openly espousing Stalinism, so an unconventional jobs program that worked raised few eyebrows.
I was going to gently explain to you something along the lines of "tone down the rhetoric if you want the people here to take your political views seriously," but then I saw you have more karma than me, so I guess you know what you're doing.
The rhetoric is toned down. The present political situation is unstable, and unstable situations are ripe for demagogues to take power. I fear that the expanded interstate commerce clause is our Treaty of Versailles. If we do not build a country to be proud of, someone will slap a few runes and occult symbols on the Flag and say "Here is something to be proud of." And the crematoria will run night and day.
I'll try for an on-topic reply this time instead of US history lesson. The way the system's intended to work is that each state regulates matters that are strictly internal to itself, and the federal government is just responsible for laws about how multiple states interact / trade with each other, and how the US deals with other countries in diplomatic or military matters. This was how it actually originally worked with the thirteen original states who viewed themselves as independent countries in a relatively loose military / economic alliance (as opposed to the modern view of states as more like large administrative districts in a single nation).
Due to all of the history I mentioned in my other comment, plus changing culture, plus the changing domestic / world economy resulting in lots more trade with faraway places, plus the expansive interpretation of the commerce clause [1], led to there being more and more things which at least touch on "interstate matters" or "foreign trade" or "foreign policy" and belong to the federal government, and less and less things that are strictly "intrastate matters" for the state government to deal with.
States still have plenty to do. First, local governments have no Constitutional authority, so city and county governments are set up by states [2]. Next, maintaining lots of roads / bridges / infrastructure, setting up laws against street crimes like shoplifting, assault, murder, etc. and maintaining police / courts / prisons to deal with them, and much of the education system are largely state responsibilities. Also, the non-criminal law system (e.g. lawsuits) is handled within the state [3].
States have their own military forces (referred to as "militia" in the Constitution, but usually called "National Guard" today) which largely consist of part-time soldiers and whose activities in modern times are mostly training and local disaster relief, but also function as reserve forces for national military activities including overseas wars [4]. States also run elections, register and regulate businesses, license and regulate various activities such as medical practice, regulate lotteries and casino gambling, and fund state universities. The federal government often offers to share some of these responsibilities, or attempts to get the state to voluntarily delegate some of its authority to the federal government (for example, a common tactic is offering federal funding for state projects, but requiring states to comply with federal rules for the use of the funds to be eligible to receive them).
[1] http://en.wikipedia.org/wiki/Wickard_v._Filburn
[2] The nation's capital, Washington, D.C., is considered not to be a part of any state, and authority for its government rests with Congress. (For the past forty years, the city's elected a mayor and city council, who do most of the actual local governing.) A continual sore point with the capital's residents is that, ironically, they have no voting representative in Congress! http://en.wikipedia.org/wiki/Washington,_D.c.
[3] Non-criminal law is called "civil law" http://en.wikipedia.org/wiki/Civil_law_(common_law) Unfortunately, the term "civil law" can also refer to the completely different concept of a legal system which is descended from Roman law. The US state of Louisiana is a former French territory and has a legal system based on civil law; the rest of the United States' legal system is based on English law (referred to as "common law"). http://en.wikipedia.org/wiki/Common_law http://en.wikipedia.org/wiki/Louisiana#Civil_law
Also, there are some non-criminal cases that can exist in federal court, for example, suing a party from a different state, or questioning the constitutionality of a law; see http://www.uscourts.gov/FederalCourts/UnderstandingtheFedera...
Of course, the Supreme Court is the highest level of appeal for all cases in both federal and state courts.
[4] http://en.wikipedia.org/wiki/National_Guard_(United_States)#...
For the most part these divisions between state and federal responsibilities, but the commerce clause, which gives the federal government to regulate commerce between the states, has been interpreted very broadly to give the federal government broad powers even over issues that are only tangentially related to interstate commerce.
The courts have gone back and forth a few times over how broad the commerce clause should be interpreted. If you are interested in the details including some interesting court cases, Wikipedia has a pretty good summary http://en.wikipedia.org/wiki/Commerce_Clause