How does Psalm 8 demonstrate contemplation on the significance of God?

While it is clear that David's Psalm 8 contemplates God's glory, it is not equally clear that the psalm leads to a complementary contemplation of man's insignificance.

1b You have set your glory above the heavens. ...4 what is man that you are mindful of him

Through interlacing ironies, the poet illuminates the majesty of God, not the insignificance of man.

9 O Lord, our Lord, how majestic is your name in all the earth!

David begins with the power of God in his majesty, above the heavens and against his foes. Immediately we confront an irony: the weakest beings there are establish God's strength.

2 Out of the mouth of babies and infants, you have established strength

The power theme continues as David witnesses the power of creation in the heavens then contrasts to the powerlessness of man. Remember, David was woefully well aware of man's powerlessness because of his own powerlessness against King Saul. David's personal powerlessness is an irony in itself since he began his career of public service by slaying the most powerful being on earth: Goliath the Philistine giant. David's contemplations ironically juxtaposing the power of God's physical creation with the powerlessness of God's human creations sends us in free-fall, like a meteor, into the next irony: the greatest of the creations is the least.

4 what is man that you are mindful of him

David confirms that this is an ironical statement with the word "Yet" that opens verse 5: "Yet you have made him ...." David has been contemplating God's perfection and contrasting it to the implied imperfection of humanity: "what is man ...?"
After pointing out the irony of God in His perfection caring about imperfection, David points out that humanity is almost as great as the angels (who themselves are imperfect considering the former rebellion in Heaven): "a little lower than the angels" (NIV). Rather than contemplating humanity's insignificance, David is contemplating humanity's significance: [paraphrase] Nonetheless, you have made imperfect man only a little less important than the holy angels, who are the ministers of the Lord. This is indeed an irony: unimportance elevated to importance.

5 Yet you have made him a little lower than the angels and crowned him with glory and honor.

A potentially controversial psalmic couplet follows. It is potentially controversial (or maybe outright controversial) because factions use it to defend a laissez-faire attitude to environmental changes and problems while other factions use it to defend heightened stewardship over the environment to try to reverse, stall or limit these changes and problems.
Remembering that, in English, the Subject leads the Verb and Object SVO, except for rhetorical or poetical effects, if we look at a loose transliteration of the lines from Hebrew to English, we see that, in English syntax, the Object of the sentence precedes the Verb and Subject and is repeated in varied words on the other side of the Subject: [transliteration] his feet under You have put all of Your hands [things] over Your works made him to have dominion (BibleHub.com).
The sense this gives is that the Subject in the transliteration, "all of Your hands [things]" ("the works of your hands"), is what is most important. Reading the loose transliteration, the sense that comes across might be expressed like this: [paraphrase] Under his feet You have put all the things of Your hands, over Your works You have given him dominion or power. This actually creates a parallel rhetorical structure (a chiasmus).
The sense that emerges is that David's wonderment at the string of ironies continues and not only continues but grows. The irony is that God, the majestic, has given power over the fruit of His work to the weakest and most powerless: the most exalted task has gone to the least exalted.
It is understandable that after these meditations, after realizations of all these ironies, after an epiphany of the power of God that grants greatness to the most lowly, David would again proclaim the majesty of God:

9 O Lord, our Lord,how majestic is your name in all the earth!

[To hear Psalm 8 sung by a Cantor: http://www.smithcreekmusic.com/Hymnology/Sound.Files/Psalm8.mp3 ]