Thank you all for your help and suggestions on the previous post. Some were mostly likely correct and the rest were entertaining, at least. I guess I am going to go with my initial instinct and say it means "to give up," which was pretty much what A. and Expat Mum said. I remember having to slap the mat when in pain in wrestling days. And I remember the ref slapping the mat when a wrestler was officially pinned (never me, though) and I remember in Judo in the military being told to clap your hands if you were choking or felt a joint being dislocated (often you weren't near the mat to slap it.) Usually I just made pitiful whimpering sounds instead of clapping my hands, as I recall. Especially during garotte training day.
None of this tells me for sure why a retired Australian building contractor would say he had to tap the mat and have someone else design his website. But close enough.