The context of Christ's statement reveals that He offers this man an opportunity to participate in His ministry and become a teacher of the way that leads to eternal life (see Luke 9:59-60). When this man says that he first had to go and bury his father, Christ tells him to let the "dead" (those unaware of spiritual matters) bury their dead, "and you go and preach the kingdom of God" (verse 60).
Was Christ telling this man not to attend his father's funeral? No. A funeral usually takes only a short time, and Christ continually shows that Christians are to have compassion and show proper respect for others. In fact, Luke provides a short account of His own reaction at meeting a funeral procession during His ministry (Luke 7:11-15).
Why, then, does Christ answer this man in such a way? He realizes that the man is only making excuses. Probably, the young man's father was an elderly man who would likely die soon. This man, leaning on his father's condition, tries to put off the responsibility that Christ offers to him. Obviously, his priorities are not based on faith and serving God first (Matthew 6:33). If he wanted to, he could find a way, without showing any disrespect, to have his father cared for and still serve God. Thus, Christ tells him to let the spiritually dead—those who were not being called (Ephesians 2:1)—continue to live their lives as seems best to them.
This account illustrates how we should not let undue concern over physical matters distract us from serving God once He has called us to His work (John 6:44, 65). The Parable of the Sower in Mark 4:14-20 relates a similar principle. As Jesus says to another disciple in Luke's account, "No one, having put his hand to the plow, and looking back, is fit for the kingdom of God" (Luke 9:62).