What does John 5:40 mean?

Those who are unwilling to accept certain ideas cannot be convinced, no matter how much evidence they are given. This is the key theme of verses 39 and 40. 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.

This, ultimately, is the reason why 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).

The Pharisees knew more about the Scriptures than anyone, heard the witness of John the Baptist, and saw Jesus' miracles. And yet, they didn't put their trust in Him. 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 really accept God, ever.