it depends which one youre talking about, but usually its a slightly thicker than usual razorblade mounted on a hinge type mechanism built into the head of the razor, you thread the thin strop through the mechanism and pull it one way and then the other the blade flips from side to side and in doing so both sides get stropped. they work ok and shave ok, for self stropping razors i prefer a rolls razor but some of the older valet auto strops are very elegant nice looking razors. i in no way profess to be an expert or even particularly knowledgeable about valet auto strops.
thanks. the rolls razor has a fairly thick blade that almost looks like a section of a straight razor (it does have a guard) that can be removed from the handle and the case contains a roller type mechanism that both hones and strops the razor. this video shows the working of them (personally i take it quite abit slower when honing mine) its abit long and there may be better example videos on youtube, this was just one of the first i saw. http://www.youtube.com/watch?v=nKJA9kTTh8k
theyre relatively inexpensive, you can pick one up in the condition on the video here (the uk) at carboot sales for around £2 fairly often. there are a few different types, oyster etc but that has more to do with the case shape and material than the mechanism and blade. there are also a few valet auto strop videos that link from that page.