I had this problem too, on my Iphone 5S. I kept getting an error message that said, "these credentials don't seem right" (or some variation of that. I spoke to an Airbnb rep for a while, and this is what finally worked:
Assuming that you have Safari on your Iphone:
1) go to Safari and click on it.
2) type airbnb.com into your URL/browser on your iphone
3) you should discover that airbnb is already open
(at this point, you can just use the site from here, or, if you prefer to resolve the problem with your app, which I highly recommend, assuming that you know your username and password)
4) click the airbnb icon in the upper-left of your iphone, to access the menu to log out, and LOG OUT.
5) Hit the round home button at the bottom of your iphone, to locate your airbnb app and enter your username and password.
This worked for me. The issue apparently was that airbnb was already open on my phone's url, and couldn't figure out how to open via the app. at the same time.
I hope that this helps.