You ask a question about which there is no certain answer.
There is broad disagreement, and the translations vary widely.
Furthermore, if we look at the parallel account in Matthew, we
see a different statement:
the days of John the Baptist until now the kingdom of heaven has
suffered violence, and men of violence take it by force.
 The law and the prophets were
until John; since then the good news of the kingdom of God is preached,
and every one enters it violently.
You can see
that the statements are very different, yet the gospels indicate a
single utterance. What did he really say, and as you ask, what
does he mean?
It is helpful to see this utterance in context. Some had thought that
John the Baptist was the Messiah, but the evidence of the gospels is
that he deferred to Jesus, and Jesus gives him high praise. John
and Jesus were two of many messianic figures in the First Century and one of the
more notorious was Judas the Galilean. This man finds mention in
Acts 5:37 in the speech by Gamaliel. There were others before and
after, but Judas is particularly important to our interest here because
of the way his history impinges on Jesus. You find information on
Judas here: www.jewishencyclopedia.
In 6 AD, as a result of the census under Quirinius, Judas the
Galilean raised a small army and rebelled. He captured Sepphoris,
the capital of Galilee, which was situated just
five miles north of Nazareth where Jesus, who was then about ten years
of age, resided with his family. It was a time of trouble and
violence with the whole territory in revolt
and many violent men seeking to take the kingdom of God by force.
you can see from the Jewish Encyclopedia, his intent was to establish
the kingdom of God with himself as regent. There were
apparently many others during this time and for two centuries
preceding, going back all the way to the Hasmoneans.
So, when we interpret this utterance, it is obvious that it
refers to the history of the times. This perception also helps to explain this utterance:
Therefore, I see the utterance as presented in Matthew 11:12 to be a
commentary on the times -- on current and immediate past events that
were big news in Israel. It is easy to see how the failures of the men
of violence to establish the kingdom could have confirmed Jesus in his
teaching of absolute non violence.
 All who came before me are thieves and robbers; but the sheep did not heed
But what about Luke 16:16? I have concluded either of two answers to this question:
1. Luke's sources erred in their recollection of what
Jesus said on this occasion.
I favor the second option because both sayings are enhanced when we put them together:
2. Both utterances (of Matthew and Luke) are genuine, and were offered at the same time. Both are accurate, but incomplete.
From the days of John the Baptist
until now the kingdom of heaven has suffered violence, and men of
violence take it by force. The law
and the prophets were until John; since then the good news of the
kingdom of God is preached, and every one enters it violently.
This makes sense when we see that the second utterance plays off the
first, so that the second would better read, "and everyone who enters it suffers violence."
Some translators put it in that or similar words. In this case,
we can see irony in the combined sayings, such as is characteristic of
Jesus: violent men seek to take the kingdom through violence, but it is
only by suffering violence that anyone enters the non violent kingdom.
Translators are at a disadvantage here because these utterances are the
only occurrences of the original Greek that indicates violence, biazw, in the NT! We have no comparisons by which to
test a translation or interpretation except from the ancient, non
Biblical Greek texts.
In conclusion, as way I understand it, Luke 16:16 refers to the
violence suffered by anyone who enters the kingdom. The meaning
of the text is then :
Men of violence, honoring the Law and the Prophets, have
been seeking to take the kingdom by force and violence until now; but
now, since John, the good news of the non violent kingdom is preached
and everyone who enters it suffers violence.
The fact that we have two separate renditions of what is apparently
the same utterance requires that one apply some such
interpretation. For this reason I seldom cite this utterance as basic
to my convictions, but if it is as I suggest, it adds to our
understanding of the non violent revolutionary, Jesus, and the non
violent kingdom of God.
It would be helpful if we could point to something more that would tend
to confirm this view. We can if we think of Jesus as entering the
kingdom when he suffered death so as to bring the kingdom to
earth. Therefore he knew that he was to enter the kingdom through
suffering the violence of the cross. He also knew that it was
necessary for everyone who would follow him into the kingdom to, in
some sense, also take up a cross:
 Then Jesus told his disciples, If any man would come after me, let him deny himself and take up his cross and follow me.
Therefore he said,
. . . every one enters it violently.