There are two threads in this passage. One is a "You can't come unless the Father draws you" thread and the other is a "bread of life" thread and the two threads are co-mingled. The entire passage is Jesus trying to speak truth to the dull of hearing. As dullness of hearing turns into hardness of heart, so Jesus' teaching becomes more parabolic in keeping with His stated reason for using parables.
As regards the "bread of life" thread, the scene starts with the feeding of the 5 thousand, followed by the same crowd that He fed meeting Him across the lake, followed by this chastisement of the crowd:
Jesus answered them and said, Verily, verily, I say unto you, Ye seek me, not because ye saw the miracles, but because ye did eat of the loaves, and were filled. Labour not for the meat which perisheth, but for that meat which endureth unto everlasting life, which the Son of man shall give unto you: for him hath God the Father sealed. - John 6:26-27
Jesus first point is directing them away from the physical to the spiritual. "You came because I filled your bellies with temporal provision but I want to fill your spirits with life."
Since Jesus has used the word labour the crowd asks "What work must we do?" and Jesus responds "Believe in me (the one God has sent)."
Why should we believe?, the crowd asks, What sign will you show us so that we can believe? Our fathers ate bread in the desert, what are you gonna do?
Jesus - Verily, verily, I say unto you, Moses gave you not that bread from heaven; but my Father giveth you the true bread from heaven. For the bread of God is he which cometh down from heaven, and giveth life unto the world.
Crowd - Give us that bread.
Jesus - I am the bread of life: he that cometh to me shall never hunger; and he that believeth on me shall never thirst.
Notice that, at this point, hunger is quenched by coming and thirst is quenched by believing. The crowd is still looking for physical food and Jesus is trying to point to spiritual provision. Next it is the Jews in the crowd who begin to murmur:
The Jews then murmured at him, because he said, I am the bread which came down from heaven. And they said, Is not this Jesus, the son of Joseph, whose father and mother we know? how is it then that he saith, I came down from heaven? - John 6:41-42
Notice that the Jews problem now is not the bread thing but the origin of Jesus. They are rejecting who He is and where He is from which was the whole point of the feeding of the crowd in the first place.
Jesus - Don't murmur. If you won't respond to the Father's drawing you will not understand. I am the bread of life and if you believe in me you have everlasting life. (vs. 43-48)
Now Jesus begins speaking parabolically to the Jews who have begun murmuring against what has been intended to draw them to the truth.
Jesus - Your fathers ate bread and they are dead. I am the bread you can eat and live forever. My body is the true manna from heaven. (vs. 49-51)
Jews - You want us to eat your flesh?!? (vs. 52)
Jesus - If you don't eat my flesh and drink my blood you have no life in you. (vs. 53-58)
It is important to recognize that the crowd (Jews included) were first instructed to come and believe for life and that the eating and drinking metaphors were introduced to those who were already in the process of rejecting Jesus' self revelation.
Later his disciples (some of whom did not believe (v. 64)) express concern over the hard teaching and Jesus says "Really, how offended are you going to be when I go back where I came from. There is no profit in the flesh. If you're not being drawn by my Father you can't come to me" (vs.62-65). At this point many disciples leave.
The entire passage is one of tension between the fleshly focus upon the physical and Jesus trying to use the physical to spotlight the spiritual. That was the purpose of the feeding of the multitude and his conversation with them. It was to those in that crowd who were rejecting the authentication of who He is that he used the eating and drinking language.
And so it is today those same, amongst the crowd who cannot quite reach that place where Jesus is actually, completely sufficient in all aspects and in every way, who must position physical religious requirements within hierarchical limitations accompanied by strange teachings which detract from the simplicity that is in Christ.
Come and believe is the means of salvation for those with ears to hear.
Eat my flesh and drink my blood is the parable to those without.