I think this blog post is a bit misleading: https://blog.trello.com/trello-mobile-offline
Unfortunately it doesnt provide the offline capabilities on an iOS as I would love to have. If I go to flight mode it says "you are offline".
It seems I to go to every (!) at least once to download it and then it available. But I am doubting I have to go to all relevant boards again to sync them, before I go offline again (even in my setting "background sync" is activated).
Long story short: A real offline capability like Evernote or todoist would be awesome. And not only for iOS, also for Win / Mac ... I would even sign up for premium just to get such a feature.
Evernote is not sexy, todoist is so-so, but Trello is really cool ... unfortunately not offline :-/