Certainly!
As you likely know, modern-day TMs and HMs require three components: a poké ball, the TM case (a bit of a misnomer, as the case works with HMs as well), and the disk itself. As a note, there are different types of cases. Some people prefer the tube-like universal case that can store multiple disks at once, but others prefer the older style of cube cases that can only take on one disk at a time.
With either style, there should be a slot for your poké ball. In the tube style, this slot is located on one end, in the white-colored groove. For older, cube-style cases, that groove is located beneath the numbered latch. Either way, you would begin by placing your poké ball in that groove, then replacing the cover to avoid injury via the digitization process. For tube-style TM cases, you would also need to insert the disk you wish to use into the labeled drive close to the ball groove; cube-style cases, especially first- and second-generation devices, will have the disk already built in. (This is partly why TMs were single-use up until recently. The whole thing tended to break after one attempt. Third-generation TMs switched almost completely over to the tube-style of case, but the problem with those was that the ejection process tended to damage the disk beyond repair, thanks to a rather nasty design flaw. Fourth-generation TM cases onward improved on the design so that disks are now ejectable on all models, but you likely already knew this.)
Once the poké ball and disk are inserted properly, it’s just a simple matter of turning the device on and initiating the tutoring sequence by way of pressing “yes.”
This is the part you were likely referring to. Inside the TM case is a digitizer and virtual grid, much like the one used to store your poké balls in your regional storage system—or, for that matter, to store pokémon inside each poké ball. Once the contained poké ball has been digitized and suspended within the virtual grid, the information contained on your TM or HM disk is uploaded to your pokémon’s data and integrated where appropriate (that is, within the parts that encode its battling memory). Once that’s complete, your pokémon is reintegrated into the real world, and just like that, it will know the move you were trying to teach it.
Of course, this process applies just to the most popular styles of TM cases. There is also a slightly rarer style of TM case put out by the Pokémon Cutting-Edge Technology Research Center, which is a more pocket-sized cube capable of transmitting move data via digital pulses, so all one would have to do is pull the case apart over a compatible pokémon. Unfortunately, these are single-use TMs for obvious reasons, and the researchers at the Technology Research Center are still working out a few kinks in the system related to that. Also, seeing as how, precisely, this system works is a bit of a trade secret and seeing as the Technology Research Center is also one of my many employers, I’m unfortunately not at liberty to go into more detail than is already covered by the promotional packet. (Sorry.)
Then, of course, you also have the people who prefer to use the TM and HM disks as DVDs. You see, although each disk actually does contain information on how the move is used, they also contain tutorials that break each move down in ways that are comprehensible to a human. Thus, some people prefer to read this data themselves and use it to construct their own methods of teaching pokémon those same moves—which, according to some communities, is far more effective than doing so by machine. It makes sense, largely because by teaching your pokémon manually, you can incorporate their needs and body plan into your method, which in turn allows a pokémon to “feel” a move as they learn it. In short, it makes the process more natural for them, which means they come to comprehend their new moves in terms of the way their bodies use it, rather than purely from an academic approach.
Incidentally, yes, humans who have learned the information contained within a disk well enough to teach that one move to a wide variety of pokémon—including those that might not be compatible with the move according to the TM case—are called move tutors. (Yes, there are move tutors that teach themselves the mechanics of moves that are not contained in TM or HM disks. It’s just that one of the ways some of them learn is through the disk system.) And I only bring this up because there is a gentleman in Goldenrod City who knows, at minimum, three of these moves and frequently offers them to trainers who wander near the Game Corner, and long story short, please do not give him any coins if he offers to teach your team Flamethrower, Ice Beam, or Thunderbolt. I assure you, the explanation behind how he’s able to do this is far more mundane than you would think, and frankly, I’ve told my father many times that he needs to stop scamming tourists.