Why did southern states stay in the union?

Presumably you mean the border-states - Kentucky, Missouri, Maryland and Delaware.

These were deeply-divided states that narrowly voted against joining the Confederacy. In the case of Missouri, inter-sectional violence continued throughout the war. In the case of Kentucky, a Confederate assembly was set up by an invading Confederate army, but it collapsed after they retreated. In the case of Maryland, Lincoln had (illegally) jailed the state's pro-Southern leaders, but so much Confederate sentiment remained that his assassination was actually planned in a Maryland boarding-house.

Still, all four states did stay loyal, and this was a big factor in eventual Northern victory.