Maturity is the individual experience for each card. It is separate from player experience (the experience you see while battling in the quest mode). Maturity for a card is increased by enhancing it. When the Maturity reaches 100%, the card levels up and its stats are increased. When the card reaches its maximum level, the Maturity will read MAX and will no longer be increased (unless the rarity of the card is increased via evolving).
The best way to increase Maturity effectively is to farm for cards in chapters 2 or 3, evolve them, and use them to enhance the cards you want. This is an effective tactic because of the lower stamina use for battling in those chapters, you can acquire many cards to use for enhancing and effectively level up your cards. Cards with higher maturity levels are much stronger and more likely to land a successful trade should you decide to trade them. They also are very effective for evolving/enhancing other cards due to the increase stats.