The Whig party fell apart due to some major internal policy divisions that kept the party positions static - while the voters' opinions were changing.

The Southern Whigs were mostly pro-slavery, the Northern Whigs were slavery neutral. During the 1840s more and more Northern voters were becoming anti-slavery and the Whig party wasn't really adopting the position. A few years earlier the Whig party had been hurt by being anti Westward expansion and losing out on the issue.

It finally died out when its leaders (Henry Clay and Daniel Webster) died the same year as a election. The power vacuum led to an overwhelming defeat by the Democrats. Many northern Whigs went to the new Republican party, while southern Whigs went to the short-lived American party.

The reason why the US can really only support two political parties is because of the first past the post voting system we use. Since the party with the most votes wins everything and the losers get nothing - having a two party system protects them more. If there are three parties, and two have similar platforms, it makes sense that they would form together to create a single party in order to take on the other party.