ἀπὸ τότε ἤρξατο ὁ Ἰησοῦς κηρύσσειν καὶ λέγειν μετανοεῖτε ἤγγικεν γὰρ ἡ βασιλεία τῶν οὐρανῶν
From that time began Jesus to proclaim and say, “transform your mind, for near is the kingdom of heaven.” (Matthew 4:17)
The good news of the present (or near) kingdom (or reign) of heaven (or God) as the reason to transform (or convert/reform) one's mind, was the primary message of Christ as reported in the NT. This is the gospel according to Jesus, the central message of his ministry, as opposed to the gospel subsequently proclaimed about Jesus.
Almost all scholars equate the “kingdom of God” and the “kingdom of heaven.” Usage varies by evangelist not by context. Matthew uses "heaven," for example.
This “kingdom” is proclaimed in contrast to the Roman Imperium and its client rulers, and by extension can be seen in contrast to human rule in general. People usually orient themselves to the current human worldly system, the way things are done, how to "get ahead" or at least to "get on" in the world. It's "the way things are," "how things are done," or even the system one might work to change from within.
The use of "heaven" in this context is in contrast to the system of Roman Imperium an ordering or system beyond the world rather than one of the world. Ouranos, "heaven" primarily means, "the heavens," as in the dome or vault of the sky. Though it can also mean the sky-abode of the gods/God. This points to the transcendent nature of this alternative kingdom, its un- or other-wordliness.
Proclaiming the current kingdom of God is a call to change not only pragmatic allegiance, but as the call to transform one's mind (metanoia) points to, a fundamental change in orientation. It is a call to comport oneself to the presence of a relationship to God and what that means in regards to one another, to live as one who is truly and wholly a citizen of that kingdom, rather than to live in a kingdom of human rulers. Such a change is a transformation, and can be considered to be a state of being, or a result of inner psycho-spiritual development.
Beyond the proclamation of this "good news," Jesus' teachings about the kingdom have the quality of the unexpected, the unworldly (non-pragmatic), and the seemingly contradictory (like a Zen koan). In this “kingdom” the seemingly fundamental acts of asserting one's rights, maintaining one's place in society, and stratagems to remain safe from misfortune are alien—the kingdom of heaven does not work that way.
The kingdom of heaven runs counter to much of accepted human psychology. There have been many variations on experiments of our concept of fairness. In one variation of these experiments, two strangers are offered one opportunity to split a sum of money. One proposes the ratio of the split, and the other only has the choice to accept what is proposed or reject it, in which case neither gets any of the money. In a strictly rational approach to this situation, the second participant should accept any split as it represents gaining money. However, if the split is significantly unfavorable to the second participant, it is rejected as unfair, resulting in loss to both. What this shows is the assumed right to half of the money (though slightly less will be accepted), leading to a feeling of loss or being cheated even when it is a net gain.
This response may seem natural in the kingdom of man, but it keeps one from entering the kingdom of heaven. Part of the metanoia is seeing through these illusionary losses to the real gain, and not just the gain for ourself, but for the other as well.