Our friend the "staring owl" shows up twice in this poem, and he does the same thing both times: sings a song that goes "Tu-whit to-who!" Besides him, we've got those birds in the second stanza that are just chilling ("brooding") in the snow. The birds in this poem serve two functions. First, that owl, while charming, is also a bit scary; he symbolizes death in a way (he's a very scary predator). Secondly, even though winter is bleak and dreary—deathly—there is life. The owl and those other birds are very much alive.
Line 6: We learn that the "staring owl" sings every night (we're curious as to what he's staring at). The "staring" here seems like a euphemism, a nicer way of suggesting that the owl is looking for something to kill and eat. He symbolizes both death and the circle of life, or rather the food chain (it's natural for him to hunt).
Line 7: While the owl's song may be "merry" to people, it probably wouldn't be very merry to other animals that might be the owl's prey (mice, cute little bunnies). For that reason, there is definitely some irony going on here.
Line 8: These words are supposed to sound like the owl's song. Clearly, we're dealing with an example of onomatopoeia. The hilarity of the song contrasts with the owl's true, predatory purpose.
Line 12: These birds definitely symbolize life amid the desolation of winter. "Brooding" means just sitting, but it is a word that is sometime associated with laying eggs, sitting on top of eggs, and the like.
Lines 15-17: As in the first stanza, the owl symbolizes death (he's hunting) but also the circle of life (he must keep the population of mice and other animals down). The speaker uses onomatopoeia to mimic the owl's sound, ironically calling the killer's tune "merry."