Apple seems to say that this image should have no extension. But how is that possible to have a file without an extension? Does this make sense? Or did I get that wrong?
It's possible for a file to have no extension - just save it from a program that lets you specify "all types". It 'makes sense' insofar as the data is still there, but no instruction set is included for how to interpret the data: If Apple has said don't provide an extension, then don't and it should be fine.
It does make sense. Files are allowed to have no extensions. In fact, the extensions are basically a legacy form of metadata that allows the OS to identify their type without having to look inside for header information. Go ahead and rename a .PNG or .JPG file to have no extension and you should have no problems.
It works fine without an extension. When iTunes loads it out of your bundle (for ad hoc distributed apps), it peeks inside the file to figure out if it's a png or jpg.
Note that when you submit an app to the app store at the iTunesConnect web site, you don't put this image in your app bundle, but upload it separately. It should be a jpg file and must have a .jpg extension or iTunesConnect rejects it.