The Summer '16 Release added the option to check
Guest Access to the Support API on Sites. I can't find any specific Salesforce API called "Support API". What API access does this allow? Salesforce SOAP API access? Access to Remote Objects?
To ensure your guest community users can securely log cases directly through contact support, enable Guest Access to the Support API and add NewCase to the quick actions.
With Summer '16, Communities built with templates that allow guest users to log cases must enable Guest Access to the Support API and add NewCase to the quick actions or their guest users will no longer be able to log cases
There is no other documentation around the so called "Support API". Maybe it is an internal SF API?
Either way the whole point of the checkbox is to allow guest users to log a SF case through "Contact Support".
Until there is some public facing documentation, I am afraid this is all we know.