From The Gospel of John, by Francis Martin and William Wright, commenting on John 6:24-35
Earlier the crowd thought that Jesus was the promised Prophet-like-Moses (6:14–15). Now they realize that he is claiming to be even greater than this promised prophet, indeed, greater than Moses himself. They ask him for another sign so that they may see and believe in him (compare Matt 16:1–4). If Jesus is greater than Moses, they reason, he should do something even greater than the signs and wonders Moses did.
They cite Scripture and appeal to a great act associated with Moses: Our ancestors ate manna in the desert, as it is written: “He gave them bread from heaven to eat.” Their citation does not exactly match any biblical text, but it is closest to Exod 16:4, which recounts the Lord’s gift of manna to the Israelites (see also Ps 78:24; Neh 9:15). This scriptural text about the manna is at the center of the whole ensuing discourse.
Presented with this text, Jesus begins the Bread of Life Discourse. The discourse follows an established manner of Jewish preaching, which elaborates on the elements mentioned in a biblical quotation, in this case, Exod 16:4, referenced in John 6:31.11
Jesus begins his response to the crowd’s challenge with the assertion that it was not Moses who gave you the bread from heaven. As with the Samaritan woman (4:20–21), Jesus redirects the crowd’s attention from their ancestral past to the present moment: My Father gives you the true bread from heaven. Moses was not the source of the manna: it was the Lord. The Lord provided Israel with manna, and he continues his providential care by now providing true bread from heaven. The modifier “true” should not be taken to imply that the manna was false, but rather that it was incomplete or less than perfect. The manna was a genuine gift from God, and it also foreshadows the even greater care that God provides in Jesus.
The description that this bread comes down from heaven connects this bread with Jesus, who, as the Son of Man, came down from heaven (3:13, 31). This heavenly bread gives life to the world. Like the Samaritan woman who asked for water that would quench all thirst, the crowd says, Sir, give us this bread always (see 4:15). As he has done before, Jesus will now raise the discussion to a whole new level.