TL;DR: Subtitle files are almost never free to redistribute, as they almost always are a non-authorized derivative work of the original movie.
Creating a text form of the audio of an audiovisual work creates a derivative work, and so does creating a translation.
You need a license to allow you to do so. For non-open movies you don't have that license.* Some movies are published under a license that does permit derivative works. For example, The Internet's own boy was published under CC BY-NC-SA 3.0 and does allow you to make and distribute derivative works like subtitles.**
If you make a translation, you have a separate copyright to the translation itself, but it forms a derivative work of the original movie. You will have to meet the license requirements of the movie, which will almost never allow for this.
When Open Subtitles claims
These files are NOT illegal warez downloads, we only offer files that we believe we are free to redistribute.
They may be correct that they believe they are free to redistribute, but they are almost certainly wrong in that that is true.
*In most cases. In theory, you could have an agreement with the movie studio that allows you to publish subtitles under some agreement. For the scope of this question, I'm going to assume that's not the case.
** The NC (NonCommercial) clause makes it fall shy of the open definition. Whether the non-commercial clause allows for the use of the work on a website supported with advertisements is a whole discussion on its own, but making and distributing derivative works on its own is allowed.