(Three paragraph analysis, followed by quotes.)
Don is perpetually dissatisfied with life. He sabotages each success as he attains it, unable to appreciate it; the pursuit is broken and his value system is broken, where he understands and wants wealth, power, influence and adoration, but doesn't experience the happiness that's supposed to come with those things. All the money and adoration in the world won't stop him from bombing out, and he bombs out season after season. It's a recurring theme on Mad Men and a classic one of tragic heroes across plays, films and books: striving for an ideal that, when attained, creates a hole instead of filling it. From the outside we get it, even if the hero doesn't: the measure of success is broken.
By the midseason finale, Don seems to recognize this, and is beginning to value things differently. He lets his second marriage go wistfully but amicably; he takes visible pleasure in workshopping with Peggy and second-fiddling the burger pitch so Peggy can achieve her success by her own healthy measure; he is honest with himself and others about what work is worth doing and is on an upswing at his company while still significantly diminished as a player in it -- and he's okay with that. He's happy, and that happiness is coming from a different place.
Things end on a cheery note. While yet another person close to Don has died, there's no heavy cloud, and while Burt was a font of wisdom none is needed this time: no sermon, no preaching. The song doesn't teach a lesson so much as pleasantly punctuate one Don has learned himself.
Margaret Lyons at Vulture has an essay on the significance of death in Mad Men that says it better, and has this important excerpt:
Not that we need the show's creator to explain this moment, but he does: Matt Weiner says this song and dance are forcing Don to wonder, "What is the real value of success?"
In an interview with Vulture posted May 27, 2014, Weiner say:
It's within the language of the show. ... We use the camera to tell a story of internal feelings. ... Does it coincide on some meta level? I guess. The fact that Robert Morse is one of the greatest song-and-dance men of all time and there was really never a way within the fabric of the show's reality to utilize that, yeah. I always wanted him to sing on the show. Roger says, "The last thing I said to him were the words of some old song." I guess that's sort of stuck in Don's head. Don would know that song. Don doesn't know that Bert dances. But for me, I picked that song and Bert's death during the moon landing from the beginning of the season to make a statement about the fact that this has been a striving for success on Don Draper's part in a very new way.