How does one become an Affiliate Professor?
A wide range of "Affiliate" and "Visiting" titles are often used to simplify bureaucracy in a long-duration academic interaction.  For example, you may have a collaborator who frequently visits, and want to give them access to buildings, networks, and other resources so that their visits are less of a hassle for everybody.  A "Research Affiliate" or "Affiliate Professor" or such status allows the institution to officially put the person into the system so that they can be issued keys, ID cards, access permissions, etc.

The mechanism for actually doing this is typically that a sponsoring professor just calls the appropriate department or fills out a form, and it just happens.