Basically, pro-slavery Southerners like William Walker wanted to conquer lands like Cuba and Nicaragua so as to expand the territory that would be open to slavery. This idea was spelled out most famously in the Ostend Manifesto, which advocated taking Cuba by force if necessary. By this time (the 1850s) the issue of slavery seemed to be decided with the continental US. There seemed to be no room for slavery to grow. Therefore, many Southerners wanted to take more land outside the US so as to increase the power of the slave states relative to the free states.
Pro-slavery southerners wanted to expand into Nicaragua, Cuba, Brazil, and other areas to provide more lands in which slavery could expand. The U.S. was eager to keep slave and free states at an equilibrium during compromises such as the Missouri Compromise (1820) and the Compromise of 1850 to appease both sides. However, expansion into Latin America would give the slave states more power with regard to territory, and if the new territories were absorbed into the Union, the expansion would give them more political power.
Private citizens called filibusterers were involved in trying to expand slavery in Latin America. For example, William Walker led an expedition to Nicaragua in 1855 and eventually became President of the country. However, he was opposed by British businesspeople and American businessman Cornelius Vanderbilt. Walker came to antagonize his Nicaraguan allies and was ousted in 1857, and his actions caused anti-American sentiment in the region to grow. While the U.S. government at times showed support for filibusterers, American expansion was limited by conflicts between supporters and antagonists of slavery before the Civil War.