What's the use of the new Star button in the project pages?
Consider this "Star"-facility as a way to "bookmark" specific contributed modules as your "favorites". Some more details about it (from What’s new on Drupal.org? - April 2017):


  Users on Drupal.org may now star their favorite projects - making it easier to find favorite modules and themes for future projects, and giving maintainers a new dimension of feedback to judge their project's popularity. Users can find a list of the projects they've starred on the user profile. Over time we'll begin to factor the number of star's into a project's ranking in search results.


Notes:


Projects that you star-ed are shown to others also. E.g here are your current star-ed projects.
When I look at the views page (without being logged in to D.O), I don't see that "star" (as in your image), I only see the star itself, and the "35". After I log in, I do get a similar Star-image like your question here.