What does the priest say during reconciliation?

First, you ask for forgiveness and tell the priest how long since your last confession. Then, you list your sins, not out of a sense of guilt, but sorrow. The priest will ask you to make an act of contrition (which means stating how sorry you are for your sins and promising to try and do better). Then the priest will extend his hand over you and say, "Through the ministry of the Church may God grant you forgiveness and peace. I absolve you of your sins, in the name of the Father, and the Son and the Holy Spirit. Go in peace." At this point the priest might offer some good advice to help you avoid those sins in the future.

The priest acts as "in persona Christi", in the person of Christ. In other Words, he is the conduit through which God offers forgiveness. He also acts as "in persona Communio", he represents and offers forgiveness on behalf of the community because when we sin, we sin both against God, ourselves and the community.