From the description Matthew provides, it appears that Jesus was exhausted. He desired to leave the great multitudes and get some rest. The easiest way to "escape" from the people was to get in a boat (cf. 8:23) and travel to the other side of the Sea of Galilee (i.e., the east side). However, before they are able to leave the multitude, a certain scribe spoke to Jesus and proclaimed his willingness to follow the Lord wherever He went.
Jesus didn't want anyone to make this sort of commitment unless they knew what they were really affirming. This scribe needed to count the cost of discipleship (cf. Luke 14:26-33). Certainly he had heard Jesus' wonderful parables concerning the kingdom, and he, like all others, expected an earthy kingdom and earnestly desired to have a place in it. Jesus' reply was intended to correct the man's false expectations - "The Son of Man has nowhere to lay His head." Jesus informed the man that He had no physical place to call home. Those in the first century who chose to follow Him wherever He went shared in that circumstance.
Another disciple also expressed an interest in following Jesus but desired to go and bury his father first. Jesus replied - "Follow Me, and let the dead bury their own dead" (Matt. 8:22). In other words, "Let the spiritually dead bury the physically dead. You, come and follow me!"
There are some who seem to think that Jesus was uncharacteristically cruel here in not allowing the man time to go back and bury his father. Such is not the case. In Jesus' era, the dead were generally buried on the same day in which they died. Thus, if this man's father was already dead, he certainly would not have been there at that time with the multitude. It must be the case that his father was in poor health or elderly, and he believed that death was not far off. So, what this disciple really wanted to do was to go and tend to his father's needs until his death, and then he would come and follow Jesus.
Jesus' response demanded that the man prioritize. The disciple had no way of knowing if his father would live another day or another 5 years, and one who wanted to literally follow Jesus during His public ministry needed to act quickly. Jesus did not remain in any one place for very long. At that point in time, Jesus' earthly ministry wasn't going to last much more than another year. Jesus was simply telling the man that he could not do both. Thus, he needed to choose what was really important to him. Certainly man does have a responsibility to his family, but this responsibility does not supersede his duty to God. Our love for our families should be strong, but it should seem like hatred in comparison with our devotion to Almighty God (cf. Luke 14:26). Friends, we today must never postpone following Jesus until "this" or "that" happens. To do such is to foolishly choose the wrong priorities. Following Christ is something one must do now, not put off until later. Now is the only time we are guaranteed.