It's incredibly simple, really.
Just under half the country is loyally Republican and just under half the country is loyally Democrat.
This means, in any given election, about 45% or so of the country is going to always vote Republican or the way Republicans tell them to, and another 45% or so is going to always vote Democrat or the way Democrats tell them to. This leaves the remaining 10% or so of the population that is legitimately undecided to throw their lot in with whichever side makes the most convincing argument and sway the vote one way or the other.
The problem is, with the way districts are gerrymandered and rigged to basically always favor a particular party, the number of undecided voters or "issues voters" is typically grossly outweighed by the number who are clearly biased towards one party or the other in most of the country. This is not an accident. Swing voters, voters who make their choices based on the issues or based on which side made the most convincing argument are an annoyance. They force politicians to pretend to give a shit about "bi-partisanship" or "the working man" or whatever other buzzword they have to latch onto in order to win these people over. It's much easier when you redraw your district lines over and over so that the Democrats will always win here and the Republicans will always win there and all you have to do is show up and you got their vote.
Of course, this totally screws the average American.
Think about it. I live in the central valley of CA. This area always votes Republican, no matter what. Obama could come through here shitting gold coins out of his ass and the Republican nominee would still get more votes here. As a Democrat, what incentive do you have to give a shit about this area? No matter what you do, they aren't going to support you here, so why fucking bother?
On the other hand, any Republican that comes through here is greeted like a king and heralded as a champion. A Republican candidate could literally get caught with a hooker and an 8-ball in a motel room in Fresno and still beat the Democrat in any given election. As a Republican, what incentive do you have to give a shit about this area either? No matter what you do, you're going to get their vote, so why go out of your way if you don't have to?
The only people politicians give a fuck about are the 10% or so of true "swing" districts, where they never just vote for one party or the other every election. Those are the only parts of the country they work hard to campaign in, those are the only people they tailor their slogans towards, those are the only people they pander to for votes. Sure, Republicans and Democrats stop off in all the stronghold states and districts, because they need to hold fund raisers and get their asses kissed and preach to the choir a little bit to get those party line soundbytes out there with tons of enthusiastic applause behind them from the base, but really it all comes down to the 10% of the country who never fully makes up their mind until election night. That's who the politicians work on, that's where they focus their voter intimidation and disenfranchisement efforts, that's where all the corruption and glad-handing and bullshittery gets whipped up to a frenzied level. Because all they have to do is get the majority of that 10% or so on their side and they win.
Of course, once they win, we all lose. Because, again, 45% of the country is going to shit on everything they do, so fuck em, why should they bother trying to make you happy, and 45% of the country is going to love everything they do, so fuck them, why should they bother trying to make you happier? No matter what, you're either in love with the guy or you hate him, so there's no incentive for them to give a half a fuck about you, ever.
That's why two-party politics fails. Because 90% of the country is going to vote for one party only no matter what they do, and thus both parties just do whatever the fuck they want as long as they can con most of the remaining 10% into liking them best once every 4 years or so. There's no more incentive for a politician to work to impress voters who hate them than there is for them to work to impress voters who love them, their opinions aren't going to change either way. A die-hard Republican isn't going to vote for a Democrat even if a Republican politician takes a shit on his head, so what motivation does that politician have to do anything other than suck the dicks of his special interests? And vice-versa.
So yeah, when you support a two-party system, you're basically supporting a system where our elected officials either do nothing for us and we hate them for it or they do nothing for us and we love them for it, but either way, they do nothing for us and we keep electing one of the two shit heads every time.