In this difficult section of text, our Lord described the kingdom of heaven as a besieged city. The city was secure, but those who surrounded it were storming its walls and trying to force an entrance. They were trying to press themselves into the kingdom (cf. Luke 16:16). These attempts to take the kingdom by force had been happening since the days of John.
It is important to understand that the gates of Christ's kingdom were not formally opened until the day of Pentecost (cf. Acts 2). However, men, hearing that the kingdom was about to be opened, sought to enter it prematurely, and not by means of the gate God would provide (cf. Matt. 16:19). There are several examples of men who tried to "force" their own opening into God's kingdom, though God was not yet ready for the kingdom to be entered. Consider John 6:15 in particular - "Therefore when Jesus perceived that they were about to come and take Him by force to make Him king, He departed again to the mountain by Himself alone" (cf. Matt. 20:21; Luke 19:11,36-38; Acts 1:6). The people were full of preconceived ideas regarding God's kingdom. They expected it to be an earthly, physical kingdom with military power, and that is why they desired to make Jesus a physical king over them. This mentality is what Jesus was describing when He spoke of God's kingdom suffering "violence". Many sought to force their way into the kingdom and enjoy its pleasures. They were about as successful as one who impatiently seizes a bud and seeks to force it to bloom with his fingers. Some things simply cannot be forced or rushed, but that doesn't stop those who do not understand from trying.
Jesus declared in Matthew 11:13 that the former age was the work of a long series of prophets, and this series was concluded with John the baptizer. John was the Elijah who had been predicted to come (cf. Mal. 4:5,6; Matt. 17:12,13; Luke 1:17).
Jesus then uttered His frequently used phrase - "He who has ears to hear, let him hear!" (cf. Matt. 13:9,43; etc.) Jesus used this proverb when He desired to give special attention to a message His listeners would be inclined to reject. His comments about John the baptizer would certainly be distasteful to those who had rejected John's preaching and baptism. Yet, in spite of their rejection, John was the Elijah who was prophesied to come, and their rejection was no reflection upon the greatness of John as the forerunner for Christ.