"Emotional maturity" means that most of our intentions are motivated by unselfish emotions. You can't expect perfection, of course, but if we have a strong understanding from our practice that unselfish emotions lessen suffering, then we realize that we're not going to lose anything by being unselfish, except suffering. So this lessens a sense of self or "me," and gives us a maturity to open to this truth of "not-self" on deeper levels because we realize that letting go of self is not threatening at all and actually brings more happiness and peace. Letting go of selfishness adds an emotional maturity so that we're more motivated to act from unselfish emotions and be less motivated by the selfish desires and insecurities that might motivate someone whose happiness is usually dependent on external things totally. We develop an inner stability because we're developing the cause of our happiness within ourselves, from these unselfish emotions, and this brings emotional maturity.