The title calls this an "allusive" haiku, and the allusion is easy to hear: the first line, "whose woods these are," echoes the opening of Robert Frost's "Stopping by Woods on a Snowy Evening." In Frost's line, the speaker worries about who owns the woods and whether he is trespassing. Here the question gets turned around in a quiet, funny way.
A barred owl, whose call is often heard as a question, "asks / nothing of me." The owl doesn't care about property or permission. It makes no demands, and it doesn't judge the person standing in its woods. The speaker seems to find that a relief.
The mood is calm and a little humbled. There's a gentle humor in the wordplay, but also a sense of ease: being in nature without having to account for yourself. Ownership, duty and human worry fall away, and what's left is the night, the trees and a voice that expects nothing back.