It's really quite easy once you get the hang of it >w<
All you gotta do is find a hill or a rock or something, just any place where your deer is above the usual ground level.
Found it? Good!~
From here on you just run + jump continuously. The jumping is the trick, if you keep jumping you won't "fall down"
And once you've achieved lift off, you simply press escape on your keyboard to bring up the menu, forcing your deer to lie down. And it'll float in the air!~

Bringing the menu down again and still remaining airsitting is a little tricky, you gotta make the deer stand in the processline while the menu is still up, and after that make it lie down again and close the menu, I think. I'm not entirely sure on that part, but experiment around and you're sure to stumble across the right way to do it >w<

Adding to what Chickenwhite said, just press esc twice so the menu opens and closes quickly. You'll still be sitting.
To stand up then without falling down, simply press the hotkey for either Listen or Dance.

I've found 'aiming' is easiest when you use your mouse for running. To adjust your position, stand up and spam the jumping key. You'll do a hop forward, then quickly press esc+esc again.

There's also some other more accurate trick to adjust your sitting position, but I'm either not clever enough for that or my keyboard-mouse skills just suck.

You're welcome, man! >w< Have fun using trees as cushions XD

Also, thank you SO much, Unplugged, I had no idea! XD