I tried it in my extension with context.openURL, but that is blocked by Apple. This is what their documentation says:
Discussion: Each extension point determines whether to support this method, or under which conditions to support this method. In iOS 8.0, only the Today extension point supports this method.
So how do I open my app after selecting to share a file from an email? Dropbox does it and Maps.me does it, so there must be a way.
Edit: sharing extension was not the way to solve this. In the comments, Tom pointed me towards a SO post with the right stuff: How does "Open With" some app in iPhone work?