It is a reminder of His body that was broken for us - of His blood that was shed for us.
Actually, when Paul quotes this passage in I Corinthians 11:23-34, it is often pointed out in churches where I’ve been that there are four things we are being called to focus on during communion.
We are to look backward to Calvary, and remember His broken body and shed blood, and contemplate the love that would die for us!
And as we partake of it, it becomes a part of us - just as His Spirit became a part of us when we first trusted His sacrifice for our salvation - we are nourished by the bread and the cup just as His Spirit nourishes our souls.
But we are also to look forward to His return - verse 26, ye do show the Lord’s death until he come. As Jesus Himself said, I will not drink henceforth of this fruit of the vine, until that day when I drink it new with you in my Father’s kingdom - Matthew 26:29. We remember His promise that He will come again and receive us unto Himself that where He is, there we may be also, and we contemplate the love that wants to live “happily ever after” with us!
But we also look within at our own hearts - I Corinthians 11:28, But let a man examine himself, and so let him eat of that bread and drink of that cup. I Corinthians 11:31, For if we would judge ourselves, we should not be judged. This is a periodic opportunity for self-inspection - to remember whether there is anything in our own hearts that is disrupting our communion with Christ or Christians. We should repent and right any wrongs before we partake of communion, because it should reflect our unity with Christ, the Head, and the rest of His body.
But a person who omits such honest self inspection eateth and drinketh unworthily and invites the Lord’s chastening because he does not take seriously his relationship within the Lord’s body. We are to contemplate whether we are walking in love toward Christ and others.
And finally, it is a look outward toward the church - I Corinthians 11:33, Wherefore, my brethren, when ye come together to eat, tarry one for another. Earlier in the chapter, he had rebuked them for having divisions among themselves (v. 18) and not partaking of the Lord’s Supper together as one (v. 21). This goes along with the previous point about self-inspection, but the focus here is to remember those around us, and to contemplate the love of the brethren that should unite us together.
I hope this will help you as you participate in communion in your church.