The Law and the Prophets were until John; since then the good news of the kingdom of God is preached, and everyone forces his way into it.
There is a wide-ranging debate over what this passage means, but the most literal interpretation seems to be the best. That is, everyone who enters the kingdom of God forces his way into it.
To our Reformed ears, with our emphasis on grace alone, this may sound strange. If God saves, how is it that we must force our way into the kingdom? The answer is that although we are saved by nothing that we do, no one is saved who does nothing.
Take John the Baptist for example. Jesus mentions him as one who "forces his way into it." Indeed, John was saved by the grace of God alone - not by anything he had done. In fact, John is the one who points out that it is Jesus who takes away the sins of the world (John 1:29). Yet the knowledge of the kingdom of God and the saving work of Christ didn't lead John to do nothing. Instead, John was a wild man, zealous in his message of repentance, forceful in his spiritual application of kingdom truths to the very end of his life. John was saved by grace, not by any work that he did, but that didn't mean he did nothing. Instead John spent his life (and death) forcing his way into the kingdom.
Here, some may object that if they are earnest and zealous and forceful and take a great deal of pains in doing their Christian duties that they may then fall into the temptation of trusting in their own works or righteousness.
Not necessarily so, says Jonathan Edwards. Here's his response to this objection in his sermon on Luke 16:16:
There is ordinarily no kind of seekers that trust so much to what they do, as slack and dull seekers.... A more awakened conscience will not rest so quietly in moral and religious duties, as one that is less awakened. A dull seeker's conscience will be in a great measure satisfied and quited with his own works and performances; but one that is thoroughly awakened cannot be stilled or pacified with such things as these. In this way persons gain much more knowledge of themselves, and acquantiance with their own hearts, than in a negligent, slight way of seeking; for they have a gread deal more experience of themselves. It is experience of ourselves, and finding what we are, that God commonly makes use of as the means of bringing us off from all dependence on ourselves. But men never get acquantance with themselves so fast, as in the most earnest way of seeking. They that are in this way have more to engage them to think of their sins, and strictly to observe themselves, and have much more to do with their own hearts, than others. Such a one has much more experience of his own weakness, than another that does not put forth aand try his strength; and will therefore sooner see himself dead in sin.... It is therefore quite a wrong notion that some entertain, that the more they do, the more they shall depend on it. Whereas the reverse is true; the more they do, or the more thorough they are in seeking, the less will they be likely to rest in their doings, and the sooner will they see the vanity of all they do.... Those that go on in a more slight way, trust a gread deal more securely to their dull services, than he that is pressing into the kingdom of God does to his earnestness. Men's slackness in religion, and their trust in their own righteousness, strengthen and establish one another.
Edwards is suggesting that the harder we strive to force our way into the kingdom of heaven, the more we will then naturally despair of our own righteousness and works, and trust only in the righteousness of Christ. Therefore, may we continue to force our way in, and while doing so learn to rest in Christ all the more.