I would like to flesh out my comment to an answer.
The key issue is that iOS as (opposed to Android) does not intend the usage of external media. Though it seems Apple is not too stringent about since they even produce a SD reader, based on which all devices that connect to USB are constructed.
The intended use for these devices is to import photos and videos into iPhoto. This isn't a decent player for longer videos. Note - that you especially are not supposed to copy videos into the videos app. But there are plenty superior players around anyway - like VLC.
There is a number of third party apps that will offer you transfer from iOS apps to external and back. But so far the options to use data from storage directly are very limited - it's usually transfer to phone first, and then open locally. Even the very flexible VLC is unable to use external storage. Though among the examples @Monomeeth found the iExpand claims to come with an app that plays videos. Personally I would be wary, as one is stuck with the video player provided.
A bit more promising are the drives with builtin WiFi - like this Sandisk Connect Wireless Stick. This sounds rather cumbersome but the nice thing is you are freer in the selection of apps. In addition to the provided app there usually is a web interface through which media can be viewed/streamed. The way videos are provided is very commons so you can find a selection of video players.