You know how it can be fun to watch a really tear-jerking romantic comedy? Or maybe there's a sad song that makes you cry, but the crying always feels kind of good. Well, folks in the nineteenth century loved that kind of thing. They bought tons of sentimental books and poems, and read and reread the sad scenes. So when Longfellow brings up flowing human tears, he's tapping into a major theme.
Line 28: The speaker has already told us that the poem he wants is one that came gushing out of the poet's heart. Now he compares that kind of emotional poetry writing to crying tears. In this simile, the tears are like the poem, and they come out of the eyelid, which is like the heart. Kind of sappy, we know, but people really loved this kind of thing.