‘Direct Messages’ in Twitter is as important as ‘Mentions’, yet it does not enjoy the same conspicuousness to facilitate quick access as the latter in the official iOS and Android apps.
In order to get to the Direct Messages section you need to go to Me >> and then Direct Messages. If you are new to the app, you might not be able to find it at all.
Good thing there are a lot of better alternative Twitter apps for iOS devices, case in point: Tweetbot, that give as much importance to Direct Messages.
As for Android users, well, good luck with finding a really good Twitter client alternative in the first place. Otherwise, you’re stuck with the official Twitter app.