This had happened in the past and the solutions then ranged from installing every old versions of Skype (2.8) to tampering with certain system settings, last year Skype released an officially supported version 6.3 for MacOS 10.5.8 but this only lasted until today.
So I tried various things and a variation of earlier solutions did it for me which I would like to share with the community. In summary: trick Skype into thinking that you are running a higher version than 6.3.
Change 3 settings in the info.plist file
|Bundle versions string, short||6.15|
Change the /etc/hosts file
This prevents that your network contacts the Skype download server.
/etc/hostsfile (with your editor of choice)
There you go.
Let's wait and see until this solution becomes obsolete.