When not to contact a professor for a PhD based on what is written on his web page? I am trying to reach out professors in my research area of interest. Some professors say on their (lab) web page that prospective grad students should submit an application for admission...

In this case, would it be beneficial to contact that professor so he/she can support my application? Does that mean that the school rules don't let the professor get involved in the process of admission? What does that mean exactly? I am asking about US schools
I assume the note on the web page is to avoid misunderstandings about the application process. In some places (not the US generally) it is the professor in charge of admissions and he/she actually hires students for the lab. If students from around the world expect that to be the case, the professor may have to deal with too many individual situations, so just makes a general statement. 

It isn't, I suspect, a statement that you should not contact him/her, but a warning that you need to go through the administrative admission procedures first before you can be considered. 

You can contact the prof for information or to express interest, of course, but be clear that you are also making a formal application through proper channels. 

The contact isn't necessary, however. There will be time for that after you are accepted to the program.