Before plunging into what Jesus is saying in these verses, it is helpful to consider what He cannot be saying if Scripture is to remain unbroken (John 10:35). In the various translations and commentaries of these verses, certain prejudices influence how scholars interpret them. The New King James translators chose the phrase "everyone is pressing into it," despite the Greek just barely supporting it. Other translations at least acknowledge the forcefulness inherent within the Greek words, rendering it as "everyone strives violently to go in" (The Amplified Bible; emphasis ours throughout) or "everyone forces his way into it" (English Standard Version). These all suggest the idea that the gospel message was so popular that everyone who heard it was beating down the doors of the Kingdom, as it were. They also contain the idea that everyone could enter the Kingdom at that time.
But both of those ideas are false.
It was not possible for everyone who heard the gospel to enter the Kingdom, no matter how vigorously one might try, and that is true even now. Only those whom God draws to the Son can enter the Kingdom (John 6:44). Matthew 16:17 shows that only by an act of the Father did Peter recognize Jesus as the Christ, the Son of the living God. Acts 13:48 says specifically that "as many as had been appointed to eternal life believed," indicating that those who have not yet been appointed to eternal life did not (and cannot) believe.
The idea that everyone hearing the gospel of the Kingdom is pushing to get in completely overlooks the specificity of God's calling and election (Romans 8:30) and the fact that He is working with only a few during this age, the firstfruits of His spiritual harvest. A person cannot truly seek the Kingdom or its King until God changes something in his mind (John 5:39-40), and simply hearing the words of the gospel does not necessarily accomplish that.