How did James Buchanan become the president of the United States?

President Franklin Pierce, elected in 1852, was so unpopular that the Democrats did not renominate him in 1856. His downfall came with the Kansas-Nebraska Act, which allowed Kansas and Nebraska's residents to vote to decide whether they would enter the union as free or slave states. This utterly enraged many Northerners, who had thought it was settled that there would never be slavery about the 36th parallel. It also led to a bloodbath as pro-slavery and anti-slavery forces rushed into Kansas to sway the vote and ended up fighting and sometimes killing each other.
Buchanan got his chance, therefore, because of Pierce's disastrous presidency. Buchanan had a long career as a politician and a diplomat, including serving as a US senator. During the Democratic nominating convention in 1856, this experience held him in good stead. He was considered steady and a potential unifier who could appeal to the North and also bring in Southern votes. He won the nomination on the seventeenth ballot and won the election by carrying the South.