Dear Ron Howard,
The sources I've found seem to indicate that the renewal of covenants through the sacrament brings into effect the cleansing process of repentance (being "washed clean"):
"In partaking of the sacrament and making these commitments, Church members renew the covenant they made at baptism (see Mosiah 18:8–10; D&C 20:37). In return, the Lord renews the promised remission of sin and enables Church members to 'always have his Spirit to be with them' (D&C 20:77)." -True to the Faith
"No one lives without sin after his or her baptism, however. Without some provision for further cleansing, each of us is lost. How grateful we are that the Lord has provided a process for each baptized member of His Church to be cleansed from the soil of sin. The sacrament is an essential part of that process. We are commanded to repent of our sins, to come to the Lord with a broken heart and a contrite spirit, and to partake of the sacrament. When we renew our baptismal covenants this way, the Lord renews the cleansing effect of our baptism. We are made clean
and can always have His Spirit to be with us." -Dallin H. Oaks
, italics added
"It is essential that we renew our covenants by partaking of the sacrament. When we do this with a sincere heart, with real intent, forsaking our sins, and renewing our commitment to God, the Lord provides a way whereby sins can be forgiven from week to week. Simply eating the bread and drinking the water will not bring that forgiveness. We must prepare and then partake with a broken heart and contrite spirit. The spiritual preparation we make to partake of the sacrament is essential to receiving a remission of our sins." -Vaughn J. Featherstone
"Do you remember the feeling you had when you were baptized—that sweet, clean feeling of a pure soul, having been forgiven, washed clean through the merits of the Savior? If we partake of the sacrament worthily, we can feel that way regularly, for we renew that covenant, which includes his forgiveness." -John H. Groberg
From these quotes, I get the feeling that the covenant we make with Christ is renewed on both ends. Our cleanliness after baptism is granted by the Lord as a reward for our willingness to follow him. It is a gift given to mark a new beginning. I think what Elder Featherstone says is essential: it is not just the physical eating and drinking that cleanses us, any more than it is being immersed in water that remits our sins. It is a matter of the authority of the priesthood and the sincerity of our hearts. When we earnestly and sincerely renew our covenant to follow, he grants, just as he did before, the cleansing of his Atonement.
Something that I think is interesting is the use of the word "essential" in the quote from Elder Oaks. I think that's because when we sin even one time, we have broken the covenant we made to keep all of the commandments, and as such, we have no hold on the promise that God has made to save us. Repentance is the idea that we have to realize this, change our minds and hearts, and make the promise again, taking it one step further this time around. We take the sacrament a bit wiser than we were last week, realizing one more weakness we have and how we can do better in the coming week. Doing this weekly, we take small steps closer each week toward repentance.