"And it came to pass that they did so, and did drink of it and were filled; and they gave unto the multitude, and they did drink, and they were filled."
Partaking of the broken bread filled the disciples, and then the multitude, so again drinking the wine filled both.
What were they filled with?
Since this is an ordinance where the promised result is to "have His Spirit to be with them," (3 Nephi 18: 7) is any meaning of "filled" adequate apart from being filled with His Spirit? Clearly this ceremony is not performed to merely fill the belly. It hearkens back to His promise to those who hunger and thirst for righteousness, that they are to be filled. What the Lord presents in ceremony is the fulfillment of the earlier promise in His sermon. There is a beautiful symmetry to His sermon, His promise, the catalyst ordinance, and the reality of being "filled" which is missing from the New Testament record. In this respect, as in so many others, the Book of Mormon is the preferred, and more revealing account of the Lord's ministry.
The Lord's work is to bring about redemption for mankind. He redeems. In ceremony, and now in reality, He is working with a multitude to bring about their preparation and redemption. It is a Master Teacher, proving by His words and deeds, that He knows how to lead souls to salvation.
It is of interest that the record prepared by Mormon in such a painstaking effort takes the time to include these details for our instruction. They are intended not merely as history, but also as a guide. The plan of salvation is to be shared, so that others may follow it and themselves be "filled" with His Spirit.
Now note the prayers all refer to Christ's Spirit. This is something apart from the Holy Ghost. It is Christ's Spirit which is to "always be with them."
What does it mean to have Christ's Spirit to be with you? To guide you?
This is an important point, and worthy of some discussion by Joseph in The Lectures on Faith. If you haven't revisited that little book and that discussion for a while, you may want to do that. I've recommended that it be read once a year. The Lectures on Faith was added to the scriptures as part of the Doctrine and Covenants. It was subsequently removed without a Church vote to agree to its removal. However, the new publications adding sections were sustained, and the tradition we have is that by sustaining the new versions with additions, it implies we also sustained the deletions. Therefore, they stand removed. Despite that, it is worth another read through, much like reading any scripture, it should happen at least annually.
The Spirit of Christ is also referred to in Section 84: 45-47. It is also described using the word "light" instead of His Spirit, in Section 88: 5-13. If you read those carefully you realize that connecting with His Spirit or His light will also put you in contact with everything else governed by Him. He is the light which "enlighteneth your eyes, which is the same light that quickeneth your understandings;"
This is why Joseph commented that the closer a man draws to God the clearer his understanding will be on all subjects. Greater light means clearer perception. It involves discerning between truth and error. It allows you to see what is dark, and who is dark, and what is light and who is filled with light. The light of a person's countenance is upon them, and if the Lord's Spirit fills you then you can see others just as He sees them.
It was His Spirit that enabled the Nephites to become one. With it, Zion becomes possible. Without it, Zion is only a theory to be abused and misunderstood. Without it, Zion is a vain ambition of conspiring men. It will not come to pass. With His Spirit, Zion is inevitable, because He will bring again Zion. If you would like to see the course of this doomed people changed at least as to your part, then these teachings and following these ordinances are the means by which the Lord's Spirit becomes available to you.