Significance of American Mathematical Monthly Problems in Math Grad School Admissions
It probably depends on the committee members, but I wouldn't expect it to help much.

If you get a really clever solution published, or they publish a problem you submit, that could make a difference.  It certainly wouldn't mean as much as a research paper, but it could be viewed as the same sort of thing on a smaller scale.

Otherwise, the potential impact is not large.  Solving Monthly problems would be viewed favorably, as evidence of talent and effort, but it's not likely to mean the difference between admission and rejection.  (Still, it's worth listing on your CV.)