What is the meaning of "let the dead bury their dead"?

It is an Aramaic colloquialism, and means that the son would take care of his aging father until he dies(or other relative), which might take many years. So the would-be disciple could not follow Jesus. see:

Idioms in the Bible Explained and A Key to the Original Gospels by George M. Lamsa HarperSanFrancisco (1985) As an example from the book: 21 And another of his disciples said unto him, Lord, suffer me first to go and bury my father. meant“take care of my father until he dies.”

Jesus suggested that those not believing in Christ (the spiritually dead) should take care of the aging parent.

For more examples of Aramaic Idiom download my PDF

http://www.waeshael.com/Christianity/Waking_Jesus/Entries/2007/1/21_Aramaic_Idiom_in_the_NT_files/AramaicIdiomNT.pdf

I think Jesus' answer can be taken on two levels. On the first level he was helping the individual who asked him to let him bury his dead. Jesus' response to him was that he could have little impact on the person that was already dead and should rather focus his efforts on helping to spread the Gospel that would impact the living before their death and subsequent judgement. Paul clarified the reason for focusing on the living when he said to the Hebrews in Hebrews 9:27 "And as it is appointed unto men once to die, but after this the judgment".

The second level comes in a prophetic form. Jesus did not waste opportunity to teach us and I think he knew that in future times there would come those who would teach false doctrines spending endless hours doing meaningless ordinances such as baptism for the dead as if the dead could change their outcome by repentance after death. How much more clearly could it be said since baptism represents burial. Therefore baptism or burial with Christ was a decision for this life only and would have as little effect on the salvation of the soul of that person as the burial of their bones would ultimately affect their Eternal outcome.

The purpose of His response may have been two fold. The first purpose was to encourage the disciples to faithfully follow Him. The second purpose and perhaps more importantly, was to teach correct theology.

First-Century Jewish Burial Practices

After a body was placed in a burial cave, it was left to decompose. The family mourned for seven days. This initial mourning period was followed by a less intense 30-day period of mourning, called shloshim. However, the entire mourning period was not fully over until the flesh of the deceased had decomposed, usually about a year later. The Jerusalem Talmud states: When the flesh had wasted away, the bones were collected and placed in chests (ossuaries). On that day (the son) mourned, but the following day he was glad, because his forebears rested from judgment. The final act of mourning, the gathering of the bones into a bone box called an ossuary, was called ossilegium, or secondary burial. According to the Rabbinic sources, the decomposition of the flesh atoned for the sins of the dead person (a kind of purgatory) and the final stage of this process was gathering the bones and placing them in an ossuary (Meyers 1971: 80-85). Jesus confronts this contrary theology. Only faith in Christ's redemptive work on the cross can atone for sin, not rotting flesh or any other work or merit of our own (Heb. 9:22, 26; Acts 4:12; Eph. 2:8, 9). Jesus may have rebuked the disciple rather harshly because they were following the corrupted practice of secondary burial.

The setting of the saying:

The Gospels record two incidents where disciples approached the Lord to request a leave of absence from following Him. The first request is recorded in Matthew 8. Jesus was about to take the Twelve across the Sea of Galilee to the Decapolis city of Gadara. Chronologically, this trip is the first recorded journey of Jesus to minister in Gentile territory. Maybe the disciple hesitated, probably because he did not want to go to those Gentiles so he made an excuse: Let me first go and bury my father.

He most likely appealed to the Jewish burial practice of ossilegium, or secondary burial, which would remove him from following the Lord for up to eleven months. Jesus saw this as an excuse not to minister to the Gentiles. As a result He rebuked him with a statement of irony and challenged the disciple to follow Him. I got most of the information from LET THE DEAD BURY THEIR OWN DEAD.