The Judeans just heard that the "bread" is Jesus's flesh -- which led them to question Jesus even more! They doubted that Jesus can give his flesh to eat (6:52). The Judeans were likely thinking literally, but Jesus is using all sorts of imagery and personification in prior explanations.
In response, Jesus talked about how in addition to how eating the bread of life results in eternal life, he provides a shocker -- without "eat[ing] the flesh of the Son" and "drink[ing] His blood", the people "have no life" (6:53-55). At first glance, the statement on eating a person is a shocker, but remember that Jesus has been using figurative language and bread as a metaphor throughout the conversation. Thus, eating/drinking Jesus is NOT literal! Instead, the eating/drinking refers to the gift that we receive because of Jesus's sacrifice. Jesus talks about how his sacrifice will pave the way for eternal life. In fact, we need to count on Jesus in order to receive such life in the first place.
Additionally, eating/drinking Jesus (as in accepting Jesus in light of the sacrifice he did) not only means getting eternal life, but it also means abiding in Jesus (6:56). As suggested later, abiding in Jesus is meant to be an everyday thing, so it's implied that the eating/drinking isn't meant to be a one-shot deal. While believing in Jesus because of the sacrifice nets us eternal life, it's important for us to focus on the significance of Jesus's sacrifice in our day to day life (the renewal we get is big!).
Finally, talks about how because of the link between the Father and the Son, a person who "feeds on [Jesus] will live" (6:57). The entire process has been orchestrated by both the Father and the Son, which is why Jesus can provide life.