Why is my paper "under review" if it contains no results?

There's a good chance the journal is getting confirmation that it is indeed a new problem. The fact that it's new to you does not mean it's actually new - perhaps you've simply not seen the paper(s) that stated and maybe even solved the problem.

The journal could also be confirming if the problem is actually interesting. It's not so difficult to come up with a new problem, but coming up with an interesting new problem would be something else.

Why woudln't they review your paper? All reputable journals review all papers that they consider publishing..

The fact that there's no proof to check doesn't mean there's nothing to check. Arguably, in a mathematics paper, it means there's more to check: why should they accept your paper that doesn't prove anything? The fact that you put everything in an appendix doesn't mean it doesn't get reviewed. The journal will be publishing that appendix, so they want to know that it's OK. (Otherwise, everybody's next paper would be "Abstract: [blah blah] Introduction: See appendix." and publishing just got a whole lot easier.)