The speaker describes a dream that begins in perfect calm. They feel good, see a bright light, and walk toward it without touching it. Then the dream turns playful: they fly like a bird, hop from branch to branch, and feel no rush at all. Nothing can hurt them there, and even when they fall, they land on their feet.
That last detail sets up the joke. The speaker says they "didn't think of a cat," and then jumps straight into one's face. It's a sudden, funny drop from a peaceful, almost heavenly mood into a small disaster. The title, "All Around Teaching," seems to hint at a lesson, perhaps that feeling safe doesn't mean you can ignore what's around you. The poem never says this outright, so that's an inference. The overall mood is light, dreamy and slightly mischievous, with the speaker seeming more amused than frightened by the surprise ending.