The idea is that you are adding mass to the headstock, which increases the mechanical impedance at the nut, which reduces the rate of energy transfer from the string to the headstock, which increases the energy reflected back down the string.
If the mechanical impedance at the nut is already high (as it ideally would be), then adding a bit to the impedance won't make as big a difference, so this may be something that has a bigger effect on cheaper and/or lighter guitars and basses.
Also, I don't quite understand if and how this has as similar effect on fretted notes as it logically would on open notes, aside from adding mass to the neck overall. I would expect that the higher you play on the fretboard, the less benefit there is to a device like this.
Online reviews and reports seem to indicate a modest but noticeable effect, and bass players seem to report that they help eliminate "dead spots" on the neck, meaning frets and strings where the notes don't sustain as long as the rest of the notes.
The mechanism for eliminating dead spots seems to be that the weight of the clamp changes the resonant frequencies of the neck itself. A resonant frequency of the neck would also be a point where the mechanical impedance at the nut is relatively low, so shifting a resonant frequency to a frequency where the instrument is not played would again preserve a higher mechanical impedance.
Whether you should buy one is probably not an easy question to answer. They're not very expensive, so it could be worth a shot, but then you have the possible issue of yet another mathom in your musical junk drawer. It also seems like the better your guitar or bass is right now, the less you have to gain from a device like this, but if you have a cheap but decent instrument this might kick it up a notch. Finally, since it adds weight, it can also add discomfort and encourage neck dive. If I were considering one I'd try to see if I can try one out at a store, either on my actual guitar or on a reasonably identical model at the store.