So the idea for levels is this: Alexa gets level-ups, up to a cap, based on the amount of battle experience she has. This cap is what I call a "regional cap", because it's no stranger to raising, just to keep the player on track. Once you reach the regional cap, you can continue to gain EXP as progress to your next level, but you won't be able to level up until a regional cap increase (at that point, it'll take one more battle to level up).
What you get from leveling up is skill points that you can apply to your skills (Breathe, First Aid, Cure, etc.), which make them stronger. You do not get any stat upgrades from level-ups.