In John 6 Jesus fed the multitudes with a few loaves and fish, but then encouraged them to seek sustenance with eternal value. He declared that God had sent bread from Heaven, which would satisfy hunger and third eternally (v. 32-35). Jesus was that bread.
His audience was still thinking of physical things when He told them that they must eat His flesh and drink His blood. The hunger of which he spoke was the hunger for righteousness (Matt 5:6). Only Jesus can provide true spiritual nourishment.
We receive that nourishment by becoming a part of His body, the church, and by receiving the cleansing from sin that His blood provides(Col 1:14). To receive these things, we must come to Jesus on His terms.