Looking at the context of this passage, it becomes obvious that the people understood what Jesus was teaching. It wasn't a passage that referred to the Lord's Supper/Eucharist; rather, it was the process of absorbing the teaching, lifestyle, attitude of Jesus as Master, Lord, teacher, and friend.
Jesus is saying to all people, who are confronted with the choice of following Him, that He needs to be their EVERYTHING. Jesus is not saying that His teachings are to be one of many that people pick and choose from. Jesus is saying that it is either all of Me or none of Me.
A story is told of a man who was in the process of buying a house. The house had many rooms with keys for each room. When asked about they keys, the current owner said, "You can by the house but there are two rooms that I won't give you the keys for." The buyer said, "I want access to all the rooms or I won't buy the house."
Likewise, God wants all the keys to our "house": our thoughts, desires, and complete dedication. With God, He wants all of us or none of us. Jesus wants to have our complete focus on Him. In so doing, who we are becomes who Jesus is. We absorb Jesus. We live for Jesus. We die for Jesus. And in turn, Jesus embraces us and becomes the main subject of our lives.
"After this many of His disciples turned back and no longer walked with Him" (John 6:66).
Humble yourself and follow Jesus and He will take care of you.
The Lord's humbled . . . John A. Reeves