When I say what or that without a following s, I momentarily press the tip of my tongue against my hard palate (the ridge at the top of the mouth). When I say what's or that's, my tongue arches up to touch the hard palate; its tip may touch my upper teeth. The tongue against the palate briefly stops the flow of air before releasing it to flow between the teeth to make the /s/ sound.
The samples on the page you linked are very clearly pronounced in an English accent.
Student or Learner