What does Matthew 27:39 mean?

Crucifixion victims were intentionally subjected to public ridicule. Human beings are naturally drawn to abuse those perceived as weaker, especially if they can accuse the other person of wrong. That habit enabled other public displays of criminals, such as the stocks: locking a person's head and hands into a wooden frame, forcing them to endure laughter and abuse from those who passed by.

Executed persons were usually placed where they could be easily seen (John 19:20). Here, the accusation posted on Jesus' cross (Matthew 27:37) likely brings even more jeering. This naturally occurring torment was part of the humiliation of crucifixion. It is one of many reasons this was considered such a shameful way to die.

Matthew's account echoes fragments from the Psalms which describe the suffering and death of the Messiah:

"All who see me mock me; they make mouths at me; they wag their heads" (Psalm 22:7).
"I am an object of scorn to my accusers; when they see me, they wag their heads" (Psalm 109:25).

In this case, to "wag the head" seems to indicate a rude gesture, or a sneering movement of the face. That came along with insults, teasing, and other abuse. Their comments in the following verses show disgust at another supposed Messiah who was not even powerful enough to avoid getting killed, let alone leading the nation in overthrowing Rome.

Opinion about Jesus in Jerusalem was sharply divided (Luke 12:51; John 7:43; 9:16; 10:19). It's true that many people praised Jesus, fewer than seven days prior, during what is now called the triumphal entry (Matthew 21:8–11). And yet, many screamed at Pilate to crucify Jesus until the governor gave in and did as they said (Matthew 27:21–26). It's possible that some loved Jesus while they thought He was going to overthrow Rome, then hated Him when it became clear this was not going to happen. Other might have despised anyone who risked bringing Rome's fury down on them all.

In any case, Jesus was accomplishing in that moment exactly what He had come to do (Philippians 2:8; Matthew 16:21). His willingness to do so (Matthew 26:39) would make it possible for the sins of those who mocked Him to be forgiven (Romans 5:8–11).