For each additional level the card gains 1% in power. Nightmare for example gives a 13% Boost to Demons Attack at level one. At level five this would turn into a 17% Boost. It is a 4% increase because the card has increased four levels (level one doesn't count as a percentage). At level 10, the max skill level, Nightmare would have 22% Boost to Attack.
The process of leveling a skill is simple in principle. If a card with a skill is enhanced using cards with skills, it has a percentage possibility of gaining a skill level. The percentage possibility has three factors.
1) The rarity of the card being evolved.
2) The rarity of the cards being used as the evolver
3) The current skill level of the card.
I found the exact numbers from Ragetrades who got the information by translating it directly from Japanese sources. The far left column signifies the skill level, the top signifies the percentage gain for each card of that rarity. For example, an SSR at level two will gain .7% chance of gaining a level per High Normal card being used or 20% per SSR used.
The information that can be deduced from these graphs is as follows.
1) Cards like Dark Knight Luciella or Popstar which have a Higher Rarity but the same power level of a lower Rarity (both about High Rare-SR power levels) are actually worse than the lower Rarity counterparts. This is because leveling the skills will be very expensive in comparison.
2) A High Rare increased in level to SR is most effectively leveled by using High Normals. You can do so from start to finish most the time at a cheaper rate than Devil Queens/Treasure HRs.
3) Devil Queens are the only true cost effective way to gain levels on cards SSR on up.
4) Treasure Rare cards are a less expensive way of attaining HR enhancers.
I will note that buyers rarely see the proper value of a card with a higher skill level so if you plan on selling the cards shortly, it's better not to spend a huge amount of money enhancing the level.