@Anonymous nailed in start to finish.
I would have been a little more brief and just said, "Yeah, there's a way to exchange stays, you pay for the person you're exchanging with's place and he/she pays you for yours. Boom! Exchange complete."
This question gets me a little snarky because we get proposed by folks for exchanges every now and then and we find it rather irritating/insulting. You go look at their crummy $75/night listing in backwater/cartel infested north Mexico, and I'd never want to stay there even if it was free. Meanwhile, they're offering it "in exchange" for my $250/night place. Nope!
"Exchange" is done with currency. Airbnb brokers the "exchange" and they want a portion of that currency (to be expected).
The last dying bastion of "exchange" would be to/from someone you actually know. However, you need the listings "exchanged" to be 1-to-1 for the exchange to be "fair." And even then, as @Anonymous mentioned, the "experience didn't leave me wanting to try it again."
Why do you even want to do an exchange? Why not make your place as great as you can, get lots of money for it, and then use that money to stay where you want?