To answer this, let us look at the psalms we say in PDZ. We will examine them by order of importance as given in Shulchan Aruch OC 52:1. (The practical implication of this importance: If you are behind schedule, and have to skip part of PDZ to say Shema and Shemone Esreh with everyone else, the “least important” psalms are the first to be skipped.)
The most important paragraph of PDZ is Ashrei (Psalm 145). According to Brachot 4b, we recite every day because it contains the entire alphabet as well as the verse "Poteach et yadecha". The point of covering the entire alphabet is apparently to indicate that every possible praise is appropriate for God. "Poteach et yadecha" teaches that God's oversight extends to every single creature - implying that nobody is exempt from giving praise.
The second most important paragraph is Psalm 150, הללויה, הללו אל בקדשו. It calls on us to praise God with shofar, harp, drum, and a bunch of other musical instruments. The purpose of this appears to be similar to that of Ashrei: we are supposed to praise God in every possible way.
The third most important paragraph is Psalm 148, הללויה, הללו את השם מן השמים. It gives a list of the things we should praise God for creating: sun, moon, stars, sky, animals, weather, topography, trees, and all human beings. In short: everything in the world.
On the fourth most important level are Psalms 146, 147, and 149, which I'm guessing we say just in order to have read all of Psalms 145-150 without skipping. Vayvarech David, Hodu, Mizmor Letodah, and so on are on even lower levels.
It seems that the focus of PDZ is on Psalms 145, 148, and 150. The common element of these paragraphs is comprehensiveness - in who should praise God (everyone), what they should give praise for (everything), and how (in every way). Mentioning "who" impresses on us that we must praise God (not someone else), and "how" teaches that we must do it now (not wait for a "better" opportunity). The point of "what" is apparently to supply the content of our praise.
Alternatively, a different motivation for mention "what" comes from the Rambam (Hilchot Yesodei Hatorah 2:1-2):
What is the way to loving and fearing Him? When a person contemplates his actions, and sees the great wonders and perceives in his wisdom that they have no set value or limit, he immediately loves and praises and glorifies, and develops a great desire to know [God's] great name, as David said "My soul thirsts for God, for the living God". And when he thinks of these things, he immediately recoils and fears, and knows that he is a small lowly dark creation, standing with [his] small and limited mind before the Perfect Mind. As David said: "When I see your heavens... what is man, that You should take account of him?" Accordingly, I [will now] explain the main principles of [God's] creation...
The idea here is that upon realizing the wondrousness of God's creation, a person will be filled with awe and impelled to praise God for the creation. The point of mentioning "what", then, is to inspire us to give praise.
The upshot of all this is that PDZ consists not so much of actual praise, but of verses discussing praise. (Hence the name - PDZ, not just "zimra".) PDZ is not directed at God - it's directed at you. Its point is to motivate you to give the actual praise that which is the main content of the Shema blessings.
One practical implication of this is the assumption in many halachic texts that PDZ is said at home, not in synagogue. Since it is not actually part of the prayer, you can technically say it anywhere. But it must be said as close as possible to the prayer, so that the prayer is inspired by it.
(Mostly based on this.)