Since this requires you to use the app, will the expansions the FFG puts out come with a code to download the DLC for the app for free, or will you have to pay for it for the app, too?

I hope it comes with a code, if FFG charges extra its like they are charging twice.

The price of the app is included in the game's price, and the price of any "DLC" will be included in the expansion's price.
The app is an integral part of the game, you pay for the whole thing when you buy the box.

As he said

^ What they said ^

If you buy a physical expansion you will not have to pay any additional money to have that product "activated" in the app. Honestly, you probably won't even have to enter any kind of code. I imagine the app will be updated automatically with each new expansion and it will just be a matter of selecting which expansions you will be playing with in that game. After all, you couldn't really cheat and add expansions you had not purchased as it would require cards and components that you didn't have. It could even add some of the benefits of the expansions to your games as a marketing tool to get you interested in buying more products in the line.

Only reason I see them locking physical expansion content behind a code is if they're worried about people proxying components....

This will likely be the end of the Print One Demand expansions, however, as they will just sell you that content through the app.