I'm having the exact same problem. Just to confirm, redownloading and reinstalling the app did not fix the issue for me. Exact same problem persists. I have checked wifi connection, cleared memory, cleared some room (just in case though I have 10GB free now, so that's not it) and rebooted iPad. None of these fix the problem.
To other posters, if you select purchase again, having already bought the episode (or bundle) app store will ask you to purchase but if you confirm it will then pop up with a message to say that you have already purchased and can download the content for free. You can't and won't be charged twice so don't worry
We need Telltale to offer a solution here, have tried absolutely everything on my end. Recent app reviews in the app store are all saying they're having issues so it seems to be a common problem... Looks pretty bad for telltale across all platforms though by the looks of this forum. Surely it isn't that hard to make an app such as this work, given their experience in episodic story games!