First of all, lets be clear. In praying the rosary you are NOT worshipping Mary. You are simply asking Mary to pray to Jesus for you. We ask those on earth to pray for us all the time. Ask yourself, how much more powerful would it be to ask Jesus's own mother kneedling before Him at His feet to pray to Him for you and your prayer requests? He loves her like any son loves his mother, and yet more perfectly than any son could love his mother. If it is the will of God, he will not deny her requests.
St. Louis De Montfort in both, “The Secret of the Rosary” and “True Devotion to Mary,” may do a better job of explaining how Mary doesn’t work to further herself or compete with Jesus. In short he explains, she gives everything to Him. She only desires us to be closer to her Son. Mary would never keep anything for herself. It is like she takes our prayers and wraps them up in the most beautiful wrapping paper and gives them faithfully straight to God. All that the Blessed Virgin Mary wants is for us to know, love, and serve God. Even when St. Elizabeth tried to praise the Blessed Virgin, she replied humbly, “My soul proclaims the greatness of the Lord; my spirit rejoices in God my Savior, for He has looked upon His handmaid’s lowliness,” (Luke 1:45-48).
St. Bernard makes an even simpler matter of why we can choose to go through Mary instead of straight to God— God established Mary in this role in the first place. In other words, since God hand-picked the Virgin Mary as the means of bringing His Only Son into the world, it is fitting that we imitate His example by sending prayers through her back to Him. St. Bernard said, “That grace should return to the Giver through the same channel through which it came.”
 “True Devotion to Mary” by St. Louis De Montfort pg 90
Why not go straight to God I’ve been asked? God, Jesus, and the Holy Spirit (one in three persons) like humility. It’s a trait that you can find stressed all through the Bible. One of many examples can be found in James 4:6, “God resists the proud, but gives grace to the humble.” In going through Mary to God, we submit ourselves and our prayers to God humbly acknowledging our sinfulness. So in praying this way we show the virtue of humility. It as if we are saying, I'm just a humble sinner, so I will ask your Mother, sinless & sitting beside you, to submit my prayers to you as she is more worthy than I to speak to you.
I used to worry that in praying through Mary I'd be taking prayers away from God and putting too much emphasis on Mary or the Hail Mary prayer. I’ve found that’s just not true. If you are meditating on the mysteries of the rosary correctly, the rosary can’t take anything away from Jesus. Jesus is at the center of the rosary, as is the Gospel message.
As a mother, I’ve found meditating on the mysteries, through the eyes of Mary, is even more powerful. I imagine being Mary and watching Our Lord Jesus from his birth through agonizing death. What an awe inspiring event must it have been to have witnessed Jesus ascending into Heaven? Seeing Christ through the eyes for Mary makes Jesus seem more real and personal. It deepens and strengthens my relationship with Him. It makes Him all the more my Lord, and it’s a great benefit having His mother as a good friend who helps me worship Him more perfectly. It is like she takes my hand and leads me right to her Son.