What does John 5:40 mean?

Those unwilling to accept certain ideas cannot be convinced, no matter how much evidence they are given. This is a key theme of this passage (John 5:39). Jesus is pointing out that the religious leaders who criticize Him (John 5:18) are not suffering from a lack of knowledge. The evidence is there, especially in the Scriptures. Jesus has also pointed out the evidence of human testimony (John 5:33) and His own miracles (John 5:36). The Jewish authorities are not rejecting Jesus because they don't have enough evidence. They are rejecting the evidence because they will not believe in Jesus. By an act of will, they "refuse" to submit to truth.

God draws a distinction between "knowing" and "trusting." The Israelites saw God perform amazing miracles in person and still disobeyed (Numbers 14:20–23; Nehemiah 9:16–17; Psalm 78:11–12). Thomas heard Jesus predict His own resurrection and still doubted what his friends told him (John 20:24–29). Judas was there to see all of Jesus' teachings and miracles and still rejected Him (Matthew 26:24–25).

God does not offer "proof" to most people because that proof is ultimately irrelevant. Those who claim they will only accept God when He has been "proven" to them won't accept God, ever (Luke 16:31). The Pharisees knew more about the Scriptures than anyone. They heard the witness of John the Baptist, from John himself. They saw Jesus' miracles with their own eyes. Yet, they didn't put their trust in Him.