The word "trips" perfectly describes the trochee. The movement from something long to something short, or from something stressed to something less stressed, is kind of like a little stumble.
From long to long in solemn sort Slow Spondee stalks, strong foot! […] (2-3)
Notice how many times the letter "s" occurs in these lines (solemn, sort, slow, spondee, stalks, strong). The repetition of the letter in six straight words mimics the repetitive nature of the spondee, which contains two long syllables in a row. The lines themselves are monotonous and slow, just like the spondee.
Iambics march from short to long. (5)
The speaker says, "iambics march." The word "march" reminds us of things that are highly regimented, like armies, marching bands, that sort of thing. The speaker implies that the iamb is very regular, almost levelheaded. It's just right – neither too fast nor too slow. Is this a key to the prevalence of the iamb in English poetry?
First and last being long, middle short, Amphimacer Strikes his thundering hoofs like a proud high-bred Racer. (9-10)
The image of a "high-bred Racer" totally fits the amphimacer. Like a "thundering" "racer," the lines themselves "race" – and loudly. Line 9, for example, doesn't even bother with real verbs. The speaker just says "middle short" and "first and last being long" rather than, say, "the middle is short" or "the first and last are long."
If Derwent be innocent, steady, and wise, And delight in the things of earth, water, and skies; (11-12)
The speaker says the future poet, his son Derwent, must be a lot of things; the emphasis on "delight" in particular is important. The poet cannot just "like" nature or "enjoy" it. He must "delight" in it, which suggests something more joyous than any old pleasure.
With sound sense in his brains, may make Derwent a poet – (14)
The phrase "sound sense" doesn't just mean that the speaker has "good sense," but also "a sense of sound." So in order to be a great poet, you've gotta get the rhythms down first.
May crown him with fame, and must win him the love Of his father on earth and his father above. (15-16)
The phrase "may crown" is similar to "may make" (14) in that, in both cases, it's not clear what or who is doing the "making." Sure, with a little work we can say "tender warmth" is doing the making, but the strangeness of a verb ("make") without a noun is an important effect. It implies that we can't quite know what "makes" someone a poet. It's just too darn complicated.