Why do Catholics put so much faith in the Virgin Mary?
One word: Mother
She is the Mother of the LORD, [Our Lady], and Our Mother too.
giving Mary praise
Mary goes to visit Elizabeth. She enters house of Zechari′ah and greets Elizabeth. As soon as Elizabeth hears the greeting of Mary, the babe in her womb leaps; and Elizabeth filled with the Holy Spirit, God, exclaims a loud cry, “Blessed are you among women, and blessed is the fruit of your womb! And why is this granted me, that the mother of my LORD should come to me? [cf. Lk 1:39-43]
After Elizabeth speaks, Mary pours forth her Song of Praise:
“My soul magnifies the LORD,
and my spirit rejoices in God my Savior,
for he has regarded the low estate of his handmaiden.
For behold, henceforth all generations will call me blessed;
for he who is mighty has done great things for me, and holy is his
name. [...] [cf. Lk 1:46-55]
Catholics can do no more than what Elizabeth did after being inspired by the Holy Spirit. All those who praise Mary also fulfill Mary's own prophecy in her Canticle.
Then prayers like the Hail Mary and the praises in Litany of the Holy Rosary are a natural product of her children and the Church.
The exchange between Mother and Son at the Wedding at Cana sums it all:
When the wine failed, the mother of Jesus said to him, “They have no wine.” And Jesus said to her, “O woman, what have you to do with me? My hour has not yet come.” His mother said to the servants, “Do whatever he tells you.” Now six stone jars were standing there, for the Jewish rites of purification, each holding twenty or thirty gallons. Jesus said to them, “Fill the jars with water.” And they filled them up to the brim. He said to them, “Now draw some out, and take it to the steward of the feast.” So they took it. When the steward of the feast tasted the water now become wine, and did not know where it came from (though the servants who had drawn the water knew), the steward of the feast called the bridegroom and said to him, “Every man serves the good wine first; and when men have drunk freely, then the poor wine; but you have kept the good wine until now.” This, the first of his signs, Jesus did at Cana in Galilee, and manifested his glory; and his disciples believed in him. [cf. Jn 2:3-11]
This passage web-searched yields a treasure trove in the Catholic heritage as the Google Search: wedding feast of cana catholic shows.
Picking an illustrative paragraph in one: Life of Mary (XIV): Wedding Feast at Cana | Opus Dei:
Mary’s trust inaugurates her Son’s messianic mission. She precedes the disciples in faith, who come to believe in Jesus after the miracle occurs. Thus our Lady assists her Son in the first moment of the formation of Christ’s new family. The Evangelist seems to suggest this when he brings his account to a close: After this he went down to Capharnaum with his Mother, his brethren, and his disciples, and they stayed there for some days (Jn 2:12). Everything is now ready for our Lord, by announcing the Good News through his words and deeds, to begin forming the new People of God, his Church.
- Her powerful intercession is shown here. She even advances the time
Jesus and His father had set for his ministry to commence. God cannot
say no to her; Jesus works his first miracle at her request.
- She is a shortcut to Jesus [and therfore rapid spiritual progress] through her
Motherly Immaculate Heart [of Mary]: by her counsel, 'do whatever he tells you,' and because of this miracle wrought via her intercession, his disciples believed in him.
- etc., etc.
On Mary being clearly the mother of the members of Christ, please see Catechism of the Catholic Church Paragraph 6. Mary - Mother of Christ, Mother of the Church, 963 & 964
Summary, Catholic praising and seeking Mary's intercession is easier explained through her Motherhood and her dignity and all her privileges [he who is mighty has done great things for me] arise from this supreme privilege: The Divine Maternity [cf. please see the supreme reason in Ineffabilis Deus].