The saints are saints because they were constantly striving, during their lives on earth, to seek God. I think a major part of this authentic drive is humility.
According to the Catechism of the Catholic Church, the prayer - and the life - of true humility is one that says, "God, be merciful to me, a sinner!" (CCC 2613, taken from Luke 18:13). Paragraph 2779 also says that "humility makes us recognize that no one knows the Son except the Father, and no one knows the Father except the Son and anyone to whom the Son chooses to reveal Him." In other words, humility allows us to see God - in everything. It allows us to see that, while we may have gifts and talents that are admirable, we never see them as being made on our own. The beauty of humility shows us that God has a hand in everything we do and in everything we are, and that without Him, we are nothing. And it causes us to rejoice. This true humility is nurtured and grown through constant, authentic prayer.
Which brings us back to the saints. All of the saints prayed. And they all prayed well. And I've got to believe that they grew in humility through their prayer. How could you not realize how small and dependent on God you are when you spend time with Him every day? You see God as He truly is, and through his grace you see yourself as you truly are. And I think that's why the saints all thought of themselves to be the greatest of sinners. They were most in tune with who they were.
A book I'm reading right now, Toward God, says that "prayer simultaneously illuminates both the good and the evil within us." Personally, that's a major reason why I sometimes shy away from prayer. The Holy Spirit gently reminds me of how I've screwed up, and of how I've failed to love. I'm a great sinner. But prayer also has the Holy Spirit showing me how, with His help, I can leave these shortcomings in the past and become a holy and new creation. In prayer, "it is not the experience of positive or negative emotion that matters but whether we become sensitive and responsive to the action of God."
St. Paul persecuted Christians until God smacked him around and told him to be an apostle to the gentiles. St. Augustine of Hippo was an over the top, promiscuous party animal until God spoke to him through scripture. Yet even after St. Paul wrote many letters and converted countless people, even after St. Augustine became a bishop, even after all the saints offered up their lives to God, they still realized that they were sinful creatures. And yet, they rejoiced in the fact that God was able to pick them up.
That is humility, and that is the consequence of prayer.