Basically what uproar said is right. The Air works as a charger. The iPhone believes it is plugged into the wall when the case is switched to on.
The bad thing about leaving the case on all the time is that the battery of the case won't last as long meaning that instead of getting about 70% of the battery of the iPhone back you will only get about 40-50%.
So what you want to is fully charge both the case and the phone and then unplug them from the wall. Leave the case turned off until you want to recharge the phone. Don't leave it on all the time. The iPhone uses more juice when plugged into the wall constantly charging the battery and when the case is turned on, that is what the iPhone believes is going on.
So when you use the phone on the battery after a bit, flip the switch on the case, and after the phone recharges back up, then flip the switch to turn it off.
I personally ues the Juice Pack Plus as my everyday case. It is basically the same case but with a bigger battery and covers over the buttons. I use my phone until it gets to around 10% and then flip the switch. After about an hour my phone is fully charged and I turn the case off. Now your case won't fully charge your phone from dead or even 20% but it will give you about 70% of the iPhone's total battery which is fine for most people.
At least that is how it was before I switched to the Juice Pack Plus. I used an Air case prior and that was about how much I got out of it on average. I switched to the Plus due to somedays having to be away from a charger for a couple of days. The Air was more than enough to get me through a full day.
Hope that made sense.