Not from the grand old masters, Not from the bards sublime, (lines 17-18)
This poem has a really strong sense of tradition, of all of the history of poetry and poems that is behind us. This makes a lot of sense, since this was meant to be a preface to an anthology (see "In a Nutshell") full of other people's poems. Here the speaker is acknowledging the contributions of great old poets, but also setting them aside a little bit. He clearly respects these "grand old masters," he's just not in the mood to hear from them right now.
Whose distant footsteps echo Through the corridors of Time. (lines 19-20)
Here's a sort of spooky image of history. It turns all those "grand old masters" into ghost-like beings, whose footsteps we can hear far away. Longfellow imagines "Time" as "corridors" which makes us think of a giant maze of dusty old hallways. It's eerie and cool at the same time, and you get a really clear sense of the huge distance of history.
Read from some humbler poet, Whose songs gushed from his heart, (lines 25-26)
Even though history is huge and distant and echoey, old feelings can come back in a heartbeat. All it takes is a good poem. When you read a poem, according to the speaker, you can reconnect with those feelings from the past. In fact, "The Day is Done" has two different ideas about how history works. On the one hand, it can feel huge and kind of alienating (like on lines 19-20). On the other hand, in cases like this one, words from the past can feel totally comforting.
Then read from the treasured volume (line 37)
One last thought about the past. A treasured book of poems can become part of your own history, something you love and remember and pass down in your family. We're willing to bet that Longfellow was hoping his own anthology, The Waif (which is prefaced by "The Day is Done"), would become just like this, a treasured book for generations.