Meter, rhyme, alliteration, assonance - these tricks of poetic sound are all forms of repetition. I wouldn't say the effect comes strictly from the repetition. Deviation from pattern packs a punch too.
You can see why these patterns would be an aid to memory. You can see why verse is easier to recall word-for-word than prose. The rhyme-scheme and the meter are like a cyclic redundancy check. If you remember the right meaning but the wrong word then frequently the wrong word doesn't quite fit the meter or rhyme scheme. It literally doesn't "sound" right.
But why these repetitive patterns should aid the stirring of emotion, that's a puzzle. The brain loves patterns, of course. These things are neurocharms, perhaps, feeding the patternlust. But how bare are mere words as compared to words with music. How bare is the pure music of words, these repetitions. But somehow we tune into them, and are touched.
How do we hear
in mere words
a kind of music?
No tune at all,
but this pulsing call
can move us.