Henry Wadsworth Longfellow is an American poet of the Romantic era. Romanticism was a great period in literature because it focused on being positive and living life in the moment. There is a lot of emphasis on nature and a reverence for nature. Emotion is big with romantics as well, since they are reacting against the cold logic of the Age of Reason, which came just before them.
Your line "let the dead past bury its dead" is a reflection on the live in the moment attitude. I'm sure that at one point you've had a literature teacher say the words "carpe diem." It means "seize the day." Longfellow is saying exactly that. He wants his reader to stop worrying about events of the past. Don't worry about those men and women of yesterday. Don't worry about studying those past events. He wants his reader to get up and go do something. Live and live now! Let the past be in the past.
There are other lines in the poem that encompass this attitude as well. "Life is real! Life is earnest/And the grave is not the goal;" gives an excitement to life and doing something with life . . . now. Longfellow says that "time is fleeting." Again, do it now. "Let us, then, be up and doing," appears in the last stanza and again is Longfellow pushing his carpe diem attitude.