The apps requires full access to the phone so it can do the things it does. Files Management, Text Messages, running the browser, controlling settings... among other things.
They just haven't done it... but they *COULD* if they wanted to.
Just like President Obama COULD us drones against US Citizens engaged in combat or terrorist acts towards the US Gov't.
Does that mean that Obama will? Perhaps, but only under emergency situations.
Does that mean AirDroid will take your data? Very very very highly doubtful.