What are the theological implications of "pray for us sinners, now and at the hour of our death"? Hail Mary, full of grace. Our Lord is with thee. Blessed art thou
  among women, and blessed is the fruit of thy womb, Jesus. Holy Mary,
  Mother of God, pray for us sinners, now and at the hour of our death.
  Amen.


From a Catholic and/or Anglican perspective, how does this affect death and its related rites?
The point of this passage is that the petitioner is asking for Mary, Theotokos, to pray to her Son, Jesus Christ, to assist them at that time, and at their death. It is hoped that her prayer and support will assist with the life of the petitioner, and assist them with grace to meet the Lord with final perseverance and strength.

While the process of death itself can be easy, sometimes what leads up to it can be horrifying, violent, brutal, painful, or otherwise tortuous. It is the hope of believers that at this time of trial, we will not fall away from Our Lord, but remain faithful to him. It is as if we are asking Mary to help us so that if we are faced with trials or pain or even torture, we will not forsake or speak against God.

Even Pope Francis has said something to this effect: "If anything should happen to me, I have told the Lord, I ask you only to give me the grace that it doesn't hurt because I am not courageous when confronted with pain. I'm very timid." 

Given the history of Christianity, and when considering the number of individuals who have been persecuted, tortured, and murdered for their faith in Christ, it makes sense that one of our key shared prayers would ask this for ourselves and all our brothers and sisters in Christ through Mary.