Apple have informed us that updating your device to iOS 9.2 or later should solve any problems with downloading the dictionaries. Please update to iOS 9.2 or later and see if this resolves your issue.
If you continue to experience difficulties, we would suggest you try force quitting the app, restarting your device, and deleting and reinstalling the app. Instructions can be found here: https://support.apple.com/en-gb/HT201398
You could also try signing out of your iTunes/App Store account and then signing back in again - to do this, go to Settings > iTunes & App Stores and tap on your Apple ID to sign out, then tap Sign In.
Please note that you will need to be connected to Wi-Fi in order to download the dictionaries - you will not be able to download them using mobile/cellular data.