Matthew 26

As I read Matthew's record of the last supper, as the Savior introduced the sacrament -

"...he took the cup, and gave thanks, and gave it to them, saying, Drink ye all of it;
28 For this is my blood of the new testament, which is shed for many for the remission of sins."  (New Testament | Matthew 26:27 - 28)

I noted that He instructed His disciples to "Drink ye all of it".  To me, that would suggest that when we partake of the sacrament, we drink all of it, symbolizing that we are willing to fully accept His teachings, doctrine, gospel, and His atonement.  We enter into a new testament or covenant to do so.

The Savior specifically teaches that it is through His blood that was shed for the remission of sins.  In my mind,  making this covenant and partaking of the sacrament is the means of receiving a remission of sins.

Some think his blood was only shed on Calvary.  But in Gethsemane -

"...being in an agony he prayed more earnestly: and his sweat was as it were great drops of blood falling down to the ground."  (New Testament | Luke 22:44)

Obviously, He shed His blood in Gethsemane as He suffered in agony.  David J. Ridges explained -

“Gethsemane” means “oil press.” There is significant symbolism here. The Jews put olives into bags made of mesh fabric and placed them in a press to squeeze olive oil out of them. The first pressings yielded pure olive oil which was prized for many uses, including healing and giving light in lanterns. In fact, we consecrate it and use it to administer to the sick. The last pressing of the olives, under the tremendous pressure of additional weights added to the press, yielded a bitter, red liquid which can remind us of blood and the “bitter cup” which the Savior partook of. Symbolically, the Savior is going into the “oil press” (Gethsemane) to submit to the “pressure” of all our sins which will “squeeze” His blood out in order that we might have the healing “oil” of the Atonement to heal us from our sins." (Ridges, David J. (2010-10-01). The New Testament Made Easier, Part 1 (The Gospel Studies Series)