The speaker does something unexpected: he sings to his own beard, treating it almost like a companion or a small god. The poem is only a few lines, but it gives the beard a lot of character. It is a "powder" on the neck and head, a "brown thing of thorn," and, most interestingly, a "bridge" to a day that would otherwise be "abominably bare."
That last image carries the feeling. Without the beard, the day seems exposed and unpleasant. With it, the speaker can cross over into something braver. The beard works like armor or a disguise, rough and prickly, but protective. The word "altered" about his head hints that something about him has changed, and the beard helps him face that.
The mood is odd and tender, half playful and half serious. The song is a little comic, yet the speaker's need for comfort feels real.