He's got the intro material out of the way, so now the speaker switches it up a little. He starts with an invitation, asking someone to read him a poem.
It's not really clear who he's talking to, who he's asking to "come" and read. Maybe it's someone who's with him, or maybe it's us, his readers. Maybe we imagine him reaching out to us, inviting us into his world.
In any case, it's kind of neat how he writes a poem asking for a poem. That sort of mirror image is a really important part of how "The Day is Done" works.
Some simple and heartfelt lay,
Our speaker's a little bit picky about the poem that he wants read to him. He doesn't want any old thing grabbed off the shelf.
He's actually got some pretty specific requirements. He wants it to be a "simple and heartfelt lay" ("lay" is an old fashioned word for a poem, usually one that tells a story). So, nothing fancy here, he wants just a basic poem that comes from the heart.
That shall soothe this restless feeling,
Apparently the speaker hopes the poem will "soothe this restless feeling." That's the feeling of "sadness and longing" he was talking about in line 9.
We think "soothe" is a word that really sounds like what it means. In a way, it captures the mood and the sound of this whole poem. The speaker wants to be soothed, but Longfellow also wants to soothe us, to remind us how a poem can make us forget our troubles.
And banish the thoughts of day.
To "banish" something means to force it to go away. That's what our speaker wants to do with the "thoughts of day."
We bet you know what this feels like. You come home, with your head buzzing with thoughts – things you did in school or at your job. Maybe you have a routine that helps you unwind. You watch TV or you play video games, maybe you drink a cup of tea.
For our speaker, a poem really helps him relax. Think of this poem he's looking for as being like a literary bubble bath.