Hail Holy Queen - debated ad nauseum, Catholics accept this because our earthly mother the Church says it is so. We accept the Church’s teachings, not out of blind obedience, but with faith in Jesus and His words to Peter and the Apostles.
Mother of Mercy - Jesus is Mercy, Mary is His mother,
Our life - It was through Mary’s fiat that Jesus was born, It is through Jesus that we have life, Mary is our mother also, through her we have life.
Our sweetness - Webster’s dictionary - Sweet - A beloved or dear person. Mary is certainly that to Catholics
Our hope - Mary is the first recipient of Jesus’ grace. She is the hope of all generations of the things to come. We can look to her as an example of what awaits us in the Heavenly Kingdom.
To thee do we cry, poor banished children of Eve - As children of Eve, we are banished but as children of Mary, we are part of the kingdom of God.
To thee do we send up our sighs, mourning and weeping in this valley of tears - As with all children, it is to our mother we take our hurts. We count on her to be our intecessor to our angry father and our defender in all things. That is not to say that God is not merciful or that we can not go directly to Him with our sorrows, it simply means that we turn to her at times, but she always leads us to Jesus.
Turn then most gracious advocate - Catholics believe in the communion of saints, that all who have gone before us can and do pray for us, as advocates, not THE ADVOCATE,
Thine eyes of mercy towards us - Jesus is Mary’s son, He is Mercy. Just as at the wedding of Cana, Mary brought to Jesus’ attention the plight of the host, we ask her to bring our needs to His attention. To turn His eyes toward us.
After this our exile, show unto us the blessed fruit of thy womb, Jesus - Mary’s life leads us to Jesus, she does so now from heaven. May we be greeted by all who went before us and may Mary the first of believers be there as well.
O clement, O loving, O sweet Virgin Mary - Merely words of honor
Pray for us O holy Mother of God - another petition for intercession.
That we may be made worthy of the promises of Christ - another petition, mostly in regards to purgatory, wherein we are cleansed of all sin and then, purified, we may take our place at the heavenly table.
This prayer is distinctly Catholic. One must believe in the Marian doctrines. You will never be able to explain it to a non-Catholic in such a way that they will not think it blasphemous.