One word: Gerrymandering.
The math is complicated, but the end result is not. Conservatives enjoy a political grip on American society far out of proportion to their actual numbers in the population. There are several reasons for this:
1. The Constitution gives each state the same number of Senators regardless of population, which favors the smaller states, who tend to be more conservative and Republican.
2. Senate rules give a 40% minority of Senators an effective veto over legislation, making it very hard to enact any kind of progressive legislation. This is why it took decades and major crises to get laws passed ending slavery, giving workers the right to organize unions, breaking up business trusts, creating the Federal Reserve system, enacting the Civil Rights and Voting Rights acts, etc.
3. Republicans recently have committed major resources and efforts to preventing the political majority from voting by throwing up obstacles to voter registration and voting. In the past, southern conservatives (who were Democrats then) did this with poll taxes, literacy tests, and even murder (the slain civil rights workers, etc.); conservative vote suppression efforts are more legal but also more sophisticated now.
4. Gerrymandering. This has become a very big and very important piece of the strategy to impose minority rule on the majority. It’s subtle, complicated, hard to understand, and doesn’t make headline news; so many Americans aren’t even aware of it, and few understand how pernicious it is. Here’s a simple illustration: In the 2012 election, Democratic House candidates received 1.4 million more votes nationwide than did Republican House candidates, yet the GOP ended up with 234 House seats to the Democrats’ 201 House seats. In some states the GOP got most of the House seats even though a majority of voters statewide voted for Democrats. How is this possible? Gerrymandering.
The GOP, unable to win the presidency, has focused on control of governorships, state legislatures, and Congress. Getting control of the first the first two gives them control of the latter, because seizing state legislative majorities confers the power to redraw election district boundaries.
The Supreme Court has interpreted the Constitution to mean that, except for the U.S. Senate, congressional and legislative districts must be drawn to conform with the court’s Baker v. Carr “one man, one vote” rule. But even if you’re required to make all the districts approximately equal in population size, you can still play a lot of highly creative games within that parameter, and the Court has imposed few other restraints on the apportionment of political power in the United States. Consequently, district-drawing has become highly sophisticated, using computer demographic modeling, and Republicans in particular have 
grown adept at carving up states so their congressional and legislative representative will be highly unrepresentative of the state’s population as a whole — in favor of the Republican minority, of course. This shouldn’t be considered constitutional or legal, because it undercuts the principle of democracy, but our conservative-dominated partisan Supreme Court refuses to interfere with it.
You can read more about this issue here.
Image: Pennsylvania is a Democratic state, but this is what its congressional representation looks like.