The principle is simple: instead of pinching the string with the tip of the screw, it is pinched between the head of the screw and the top of the carriage to lock it in place; the string is wrapped at least 180 degrees around the body of the screw before being tightened. Wrap-Lock Headless Tuners consist of the male threaded string lock screw, the female threaded moving carriage, and a surrounding shroud (which may be part of the carriage, or part of the tuner body, or both) which ensures that the string is guided and retained under the screw head before and while the lock screw is tightened.
This configuration provides much more active surface area than the tip of a set screw, which means much less pressure is needed to exert the same amount of friction with the string. Less pressure means less turning force needs to be applied to the lock screw, so there is less likelihood of damaging the hex key while locking the string; additionally, using a regular socket screw means we use a larger hex key to begin with, so it is very unlikely to be stripped even under abusive levels of force. Since the required level of turning force is so much less, those with nimble fingers may substitute a thumbscrew for the hex-head screw, meaning that tools would not be required for string changes at all!
Given that the pressure per unit area is so much less, the carriage is less liable to being worn or embossed by the strings too. And having the string bent around in a semicircle around the screw adds additional resistance to slippage, since the stiffness of the string itself means that it acts as a hook. Furthermore, the action of tightening the screw has the tendency to pull up a small amount of slack in the string. It means the tuners can be very compact since the carriage needs only 1cm of travel, or even less.
In the designs I have made so far using the Wrap-Lock principle, the string lock screw performs a dual function: it also keeps the carriage from rotating as the tuning screw is turned. Many headless tuners that use a cylindrical barrel require an additional pin or screw to do it; eliminating this piece saves time and money, since we don’t have buy the screw, to drill and tap the hole, and cut the corresponding slot. Last of all, this regular socket screw used as the string lock is cheaper, and easier to replace (not to mention harder to lose) than a set screw.
I have already put the Wrap-Lock Tuner into use in different situations. Here they are in a 3-d printed tuning tailpiece for my headless archtop guitar “Breaking Wave”: