There are two things that get upgraded with each skill level
For Damage skills, the base dmg that the skill does gets increased with each level. It's different for each dmg skill but usually between +5% to +10% dmg increase for each level.
For Support skills, the skill cooldown gets reduced with each level. With cooldown I mean the amount of turns you would have to wait until you can use it again in battle.
I hope this helps^^