There are at least three major things that this cartoon has to tell us about the politics of its time and the 1860 election in particular.
First, it shows us the central impact of the Dred Scott case on politics in the years after that decision was handed down. The decision overturned the Missouri Compromise, thus putting the issue of slavery at the very center of American politics. This is why the caricature of Dred Scott is the one playing the fiddle; it is his tune that makes all the politicians dance.
Second, it shows us how divided the country had become over the issue of slavery. The four couples in the picture represent the four presidential candidates in the race. There were four because the Democratic Party had split into a Northern and a Southern party over the issue of slavery. Also, not all Whigs had become Republicans or Democrats. Some had formed the Constitutional Union Party, which tried to avoid the issue of slavery.
Finally, it shows how pervasive racism and nativism were at the time. The Scott character looks nothing like him and is just a caricature. Lincoln is paired with a black woman to symbolize the idea (held by enemies of the Republicans) that they were pro-black. Douglas is dancing with a terrible caricature of an Irish immigrant, complete with a cross to symbolize the Catholicism of the Irish. The fact that all of these are seen as negative things shows how racist and nativist society was at the time.,