"determine the pawn structure" => create a fixed center by releasing the tension
In general releasing the tension is a strategical mistake by white, bringing black closer to equality. White only does this if there is a concrete followup available.
"Why would one do (d4-d5)?" => (a) to play on the queenside with pawn advances or (b) to play against a hole on d5 after (b1) ...c6xd5 or (b2) d5xc6 if ...b7xc6 is not available. (Very rarely in b1 will white retake with the e4-pawn, these positions are known from the Sicilian and are generally easier for black than defending the tension in the Philidor.) Releasing the tension with d4-d5 closes the position AND it creates an opposite wings pawn game, so if white is not ahead in this game strategically, white should not do it.
"why should (Nf3-h4-f5) follow on dxe5?" => much more complicated to answer. Releasing the tension with d4xe5 opens the position, AND it worsens white's pawn structure compared to black's (trade good wPd4 vs less-good bPd6, frees bBe7), so unless white has a concrete tactical or attacking followup, white should not do it. The GM has given the briefest possible explanation: "Nf3-h4-f5". The usual white sequence is like Qd1-e2, h2-h3 (necessary against a possible ...Nd7xe5 and ...Bc8-g4), Rf1-d1 (latent pin on the d-file), black usually plays ...Qd8-c7 (away from the kingside and momentarily leaving bBe7 unguarded, also losing contact with the h4 square). Now in some positions white is ready: d4xe5 ...d6xe5 (if ...Nd7xe5 then Bc4-b3/a2 and white has an edge in central pawns) Nf3-h4 threatening Nh4-f5 and black's whole kingside will be under pressure. Black can't keep the knight out with ...g7-g6 (too weakening) and somewhat awkwardly needs to move the bNd7 (hopefully not also away from the kingside) in order to play ...Bc8xf5, which in any case gives white the bishop pair. Winning the bishop pair would be a sufficient justification for the "mistake" d4xe5.
So black needs to know what they are doing to not fall into this Nf3-h4-f5 pressure, and theory suggests specific countermeasures against Qd1-e2. Most GMs just play Rf1-e1 instead of Qd1-e2, and once Rf1-e1 happens then d4xe5 almost never leads anywhere positive.
Example game where black did not defend correctly. Chess Machine vs Danailov, Oviedo rapid 1992 https://www.365chess.com/game.php?gid=1938013 . Kevin J. O'Connell wrote "The normal move is 13...Nc5, after which black stands very well indeed." --Inside Chess (1993.03.08), pg.29
White to move. Author Bauer, Christian in his book The Philidor Files - Detailed Coverage of a Dynamic Opening commented the following regarding this (main) position:
As for white, he can determine the pawn structure by either pushing d4-d5 or playing dxe5, the latter only making sense if Nf3 -> Nh5 -> Nf5 can follow.
What did he mean by that?
1. The push d4-d5, how does it define the pawn structure strategically? Why would one do this?
2. The more intriguing question, why should said maneuver follow on dxe5? How are the related?