Revelation 13:8 tells us that Christ's death was foreordained from the foundation of the world. By extension, that means that His life in a human body was foreordained as well. He had to be given a form that could die. Hebrews 10:5 says, Therefore, when He came into the world, He said: ‘Sacrifice and offering You did not desire, but a body You have prepared for Me.' Because mankind is made of flesh and blood, Jesus likewise partook of the same things (Hebrews 2:14, ESV). God's plan necessitated that the Creator partake of the same physical life and substance that He had created.
Because flesh is mortal, death is always an aspect of human life, and Christ's life in the flesh was no different. Some scriptures focus on the death aspect of Christ's human body, and those are the ones we tend to think of first. For example, I Peter 2:24 says that Christ bore our sins in His own body on the tree. The body God prepared for Him was marred and disfigured so much at His crucifixion that He no longer looked human.
A primary effect of the sacrifice of His fleshly body is reconciliation, both between God and man as well as between men - Jews, Gentiles, and everyone else. And because of that reconciliation, there can be fellowship - communion - with God and with all others who likewise fellowship with God, which fits with the passage we saw in I Corinthians 10. When we partake of that one bread, we are reminded of the brutal end of Christ's life as a Man, which brought about reconciliation and allowed for spiritual fellowship.