When we read this poem aloud, we can almost hear the speaker on one of his long walks through woods and fields, his footsteps beating steadily on the ground to the rhythm of the poem. The air is cool and calm and the birds and bugs are chirping.
The repetition of the structure of lines three through six make us feel comfortable, but keep us moving forward, as if we had plenty of time but were walking at a decent pace. The rhyme has a similar effect: every line features an end rhyme with another (though lines 4 and 7 end in the exact same word, which is a kind of cheating). The use of rhyme creates a sense of sonic harmony, but there are enough different rhymes that we get some variety too.
In a similar way, the poem employs assonance, using rhyming vowel sounds within the lines to create chimes in the ear. For example, we get the long I sound in "sky," "die," "child," and "piety," and the long O sound, which we find in "rainbow" and in the word "so," which is repeated three times. So, it's not just the last word of each line that's tying things all together.
To really get a sense of how the sounds in this poem work together, you've got to read this aloud. Go ahead. Nobody's looking. In addition to fully appreciating the sounds at work here, if you repeat it enough times, you might find that you start to memorize it. Then, the next time you see a rainbow, you'll have a poem on hand for the occasion to impress your friends and neighbors.