What was the "white" thing put on the casket?

The "thing" is called a "pall" which means a cloak or sacred covering. It symbolizes the white garment that one receives when they are baptized into the Catholic church. It is placed on the casket at the beginning of the funeral Mass to show that death is the completion of baptism. In the 1989 Order of Christian Funerals, there is no text read while the pall is placed on the casket. At one time a prayer was read.

The reference about white cloths/robes comes from the Book of Revelation. In Revelations 7: 13-14, the saints appear in white robes that are symbol of Christian perfection. This is part of how this practice continues today.