The divisions are based on wealth, not the same as divisions based on leaders (see chapters 1-3).
The Lord's Supper means a meal "consecrated to the Lord" or "in honour of the Lord".
The reason why their meal ceases to the "the Lord's Supper" is that "each of you goes ahead with his own supper" (translated "without waiting for anybody else" is slightly inaccurate).
By humiliating the poor, they are in fact despising the church of God. It is a destruction of the meaning of the Supper beause it destroys the very unity as one body which that meal proclaims.
The whole passage (v.17-34) is about the abuse of the poor in the Lord's Supper as various paragraphs are connected by the words "for" in v.23, "therefore" in v.27, and "so then" in v.33.
The Last Supper of Jesus was a Passover meal at which he reinterpreted the bread and wine in terms of his body and blood which were soon to be given over in death on the cross.
The words "for you" are from Is 53:12, where the Suffering Servant "bore sins for many" as the substitution of his death in our place; thus the bread manifests God's deliverance of us from sin.
The phrase "in remembrance of me" or "as a memorial of me" can be interpreted two ways:
Jewish prayers in the Passover meal: "may it be recalled the memorial of the Messiah, the memorial of Jerusalem, ...may it be present before your face to obtain for us liberation, prosperity, grace, generosity, mercy, life and peace." The memorial is the recall before God of what he has already done for his people, so that he may grant them today all the benefits.
Similarly in the Bible: "Aaron will bear the names of the sons of Israel...as a continuing memorial before the Lord." (Ex 28:29) and "The angel answered (Cornelius), 'Your prayers and gifts to the poor have come up as a memorial offering before God.'" (Ac 10:4)
The words "blood of the covenant" come from Ex 24:8 where blood was sprinkled over the people to ratify the covenant. The words "new covenant" come from Jer 31:31. Thus the cup manifests the new covenant (salvation by faith in Jesus) and the renewal of that covenant.
"as often as you drink"=a regular repeated practice, not occasional or sporadic occurrence
"until he comes" reminding the Corinthians of their essentially eschatological existence.
Partaking of this meal "in an unworthy manner" points to the abuse of the poor at the Lord's Table. Older translations: "partaking it unworthily" which would then refer to the manner of the person doing the eating. Surely, reflecting on the Lord's sacrifice for us and reflecting on ourselves are worthy activities, but the context of the passage points to another interpretation.
To profane the meal as they are doing is to put themselves under the same kind of liability as those responsible for the Lord's death which the body and blood represent.
"To examine himself" is not a call for deep personal introspection to determine whether one is worthy of the Table. The content of this examination is spelt out in v.29. It means to examine themselves in terms of their attitudes toward the body (church), how they are treating others.
In some translations, the words "unworthily" and "of the Lord" (not in the original letter) were mistakenly added. If "the body" refers to the Lord's body, then it means failure to reflect on his death, as they eat. However, even though Paul uses the words "eats and drinks", only "the body" (nothing about "the blood") was mentioned because he was referring to the church as one body (10:17), that is, Paul's concern is with the problem of the rich abusing the poor.
They should recognize or discern that in the Lord's Supper, they proclaim they were one body. To fail to discern the body in this way by their abusive action is to incur God's judgment.
Paul asserts that some illnesses and deaths are the result of their violations of the "body".
The judgment is not eternal damnation, but divine discipline of temporal sickness and death.
The solution of the problem is: (a) in fellowship meals, they should "receive" ("accept", "welcome", "wait for") each other; "receive" can mean demonstrating hospitality; (b) if they want to have their private meals, they should do that at home apart from this fellowship meal.