We had a guy come into our store to check if his computer had anything bad on it - Apparently someone called him from "Microsoft", saying that they'd detected a number of viruses on his computer and wanted to take control of it to make sure, just follow these directions... So he let him.
My Dad (aged over 80, but normally pretty smart with computers) got a call from Microsoft, telling him that there was a problem with his Windows installation. It apparently wasn't legitimate and at some point or other they said they'd "deactivate" the machine if he didn't buy a valid licence...
The guy at the other end managed to soft-soap him (got him into the usual Windows log viewer, one thing he didn't know about) and even got him to the stage of allowing a remote desktop, so that Dad could open up a notepad window in anticipation of typing "I agree to pay £90", or whatever they were going to ask him to do, and probably shortly afterwards he would have been asked to type in his payment details. Thus (using proxies for the remote desktop component, probably) they could remain as anonymous even while they accrued payment details from their victims (to be used themselves, or sold on to others). But by now (although some would say it took long enough, but they probably used a wedge of credulity, ramping up the situation subtly as the call went on) Dad had gotten suspicious, said he wanted to talk to someone who knew about such things (me... not wanting to blow my own trumpet).
Long story short, I got him to make sure the machine was turned straight off, and later that night (while off-line, to make sure) I scanned and probed and otherwise checked with a number of common and obscure security tools I regularly use to make sure there was absolutely no malware installed. (Looks like they didn't get around to that, even if it had been part of their plan.)
But they even called back (I did suggest he should have told his telecoms provider, perhaps to get some evidence of where the call originated, but there's already issues there with bad tech support response) and tried to brow-beat him into buying a "proper licence". Never mind the rage one guy (transferred onto from the previous soft-soapy one, when it was apparent Dad wasn't going to be a push-over) went into regarding the possibility that he could be talking to someone else... "Who can help you? We can help you, nobody else can! You must buy a valid licence! If you do not, we will shut your computer down!"
Obviously I'm only getting bits of the story[1], but Dad's no push-over, so the initial hook must have been very persuasive, and they even had the hard-sell guy as back-up for when the target gets less cooperative, and I can imagine some people would get brow-beaten back into a state of cooperation.
[1] Like father, like son... He tends to ramble. Mind generally as sharp as a steel trap, but doesn't necessarily concentrate on the bits of information I would find most useful when passing on such information.