The verse in question appears in a section of "Matthew" that runs from 11:2 to 11:19. The passage is Jesus' commentary on the ministry of John:
Mat 11:7 As they were leaving, Jesus began to speak to the crowds
about John. "What did you go out into the wilderness to see? A reed
shaken by the wind?
The "teachable moment" occurred because John had sent some of his disciples to ask Jesus to confirm that he intended to fulfill John's expectations for the Anointed of God, including, we are led to infer, the violent expulsion of Roman rule from Jerusalem.
Mat 11:2 Now when John in prison heard about the activities of the
Messiah, he sent a message by his disciples Mat 11:3 and asked him,
"Are you the Coming One, or should we wait for someone else?"
We know that a revolutionary fervor was in the air and John announced that Jesus would right the wrong of the oppression of the Jews:
Luk 3:15 Now the people were filled with expectation, and all of them
were wondering if John was perhaps the Messiah. Luk 3:16 John
replied to all of them, "I am baptizing you with water, but one is
coming who is more powerful than I, and I'm not worthy to untie his
sandal straps. It is he who will baptize you with the Holy Spirit and
fire. Luk 3:17 His winnowing fork is in his hand to clean up his
threshing floor. He will gather the grain into his barn, but he will
burn the chaff with inextinguishable fire."
Joh 1:29 The next day John saw Jesus coming toward him and said,
"Look, the Lamb of God who [violently] takes away the sin of the
world!
Jesus' response was to show that he was fulfilling the Messianic mission but in peaceful ways:
Mat 11:4 Jesus answered them, "Go and tell John what you hear and
observe: Mat 11:5 the blind see, the lame walk, lepers are cleansed,
the deaf hear, the dead are raised, and the destitute hear the good
news. Mat 11:6 How blessed is anyone who is not offended by me!"
Jesus gives a speech about John that portrays him as not being just another "stuffed suit" but rather the mightiest of the prophets of the closing age and the civil engineer of the Messianic age:
Mat 11:7 As they were leaving, Jesus began to speak to the crowds
about John. "What did you go out into the wilderness to see? A reed
shaken by the wind? Mat 11:8 Really, what did you go out to see? A
man dressed in fancy clothes? See, those who wear fancy clothes live
in kings' houses. Mat 11:9 Really, what did you go out to see? A
prophet? Yes, I tell you, and even more than a prophet! Mat 11:10
This is the man about whom it is written, 'See, I am sending my
messenger ahead of you, who will prepare your way before you.' Mat
11:11 I tell you with certainty, among those born of women no one has
appeared who is greater than John the Baptist. Yet even the least
important person in the kingdom from heaven is greater than he.
It is at this juncture that Jesus speaks of the violent nature of John's ministry saying that the time of John's violent image of salvation was being superseded by the kinder and gentler new order:
Mat 11:12 "From the days of John the Baptist until the present,
the kingdom from heaven has been forcefully advancing, and violent
people have been attacking it, Mat 11:13 because the Law and all the
Prophets prophesied up to the time of John.
"Matthew", ever anxious to see the new build exponentially on the old suggests that the Jewish anticipation of the arrival of Elijah is misguided and they need no further than to John who came in the "spirit [speech] and power of Elijah":
http://philologos.org/__eb-lat/appen08.htm
Luk 1:13 But the angel told him, "Stop being afraid, Zechariah,
because your prayer has been heard. Your wife Elizabeth will bear you
a son, and you are to name him John. Luk 1:14 You will have great
joy, and many people will rejoice at his birth, Luk 1:15 because he
will be great in the Lord's presence. He will never drink wine or any
strong drink, and he will be filled with the Holy Spirit even before
he is born. Luk 1:16 He will bring many of Israel's descendants back
to the Lord their God. Luk 1:17 He is the one who will go before the
Lord with the spirit and power of Elijah to turn the hearts of parents
to their children and the disobedient to the wisdom of the righteous,
and to prepare the people to be ready for the Lord."
Mat 11:14 If you are willing to accept it, he is Elijah who is to
come. Mat 11:15 Let the person who has ears listen!
Mat 17:10 So the disciples asked him, "Why, then, do the scribes say
that Elijah must come first?" Mat 17:11 He answered them, "Elijah is
indeed coming and will restore all things. Mat 17:12 But I tell you
that Elijah has already come, yet people did not recognize him and
treated him just as they pleased. In the same way, the Son of Man is
going to suffer at their hands." Mat 17:13 Then the disciples
understood that he had been speaking to them about John the Baptist.
So the key to understanding Jesus' words about "violence" here it is important to consider the context which is clearly about the ministry of John. John, like all before him hoped for the arrival of the Messiah who would rid the holy People, city and temple of their oppressors and set up a Davidic throne and rule all nations with a rod of iron:
Rev_19:15 And out of his mouth goeth a sharp sword, that with it he
should smite the nations: and he shall rule them with a rod of iron:
and he treadeth the winepress of the fierceness and wrath of Almighty
God.
The "mystery of the kingdom" and the scandal is that Messiah arrives but does NOT assemble an army, evict Rome and glorify a holy Jerusalem. At least not yet.
Jesus then goes on to criticize "this generation" (his contemporaries, not the Jews) because they rejected John for his "uncultured ways" and call to grieving for sin and then rejected Jesus because of he announced a festive wedding:
Mat 11:16 "To what can I compare the people living today? They're
like little children who sit in the marketplaces and shout to each
other, Mat 11:17 'A wedding song we played for you, the dance you
simply scorned. A woeful dirge we chanted, too, but then you would not
mourn.' Mat 11:18 Because John didn't come eating or drinking, yet
people say, 'He has a demon!' Mat 11:19 The Son of Man came eating
and drinking, and they say, 'Look, a glutton and a drunk, a friend of
tax collectors and sinners!' Absolved from every act of sin, is wisdom
by her kith and kin."
Miscellaneous
I read an Anchor Bible Commentary on Revelation by, I think, Ford who said that she thought Revelation was written by John the Baptist. She reasoned that John was called "the greatest prophet" and yet he is generally eclipsed by Isaiah, Jeremiah, etc. if he didn't write anything. And the author of the Revelation is said to be named "John". I found it interesting but it seems the author has since abandoned the view though I don't know why.