The notice has a lot to do with legacy requirements in the United States to claim the copyright to a work. Up until 1989, the copyright notice was required. Today, the statements are mainly maintained to protect against "innocent infringement" which might reduce what a content owner can get in court.
What exactly do those terms entail?
That the owner stated owns all rights and you may do nothing with the content.
My biggest concern is this: by writing that, is the company claiming to own everything on the website, even potentially copyrighted user-submitted material?
That's exactly what they are doing. Depending on the terms of the specific site, content contributors generally either assign copyright to the site owner or license the content in a way that allows the site owner to do exactly what they want with it.
Site creators with the smarts or money to do it right/get someone to do it right usually state something like:
Copyright [Site Owner] and contributors.
Other sites (like this one) state specifically what they hold the copyright to:
site design / logo © 2015 Stack Exchange Inc
THIS IS NOT LEGAL ADVICE. CONSULT AN ATTORNEY REGARDING YOUR SPECIFIC SITUATION.