Start out with an iPhone app called FlightTrackPro.

To sync this app with your TripIt account, we must first login

we're taken into Safari, where we sign in through the web browser

We see an access request...

...scrolling down... we Grant Access here

Note that all the permissions are spelled out simply here.

If we grant access, Safari zooms out, bringing FlightTrackPro back into the foreground.

...and voila, with OAuth, FlightTrackPro now has access to our trips.