Pad clamps ( or key clamps ) come in many shapes and sizes, but the basic principle remains the same - their job is to hold the normally open keys closed against the tone hole when your horn is in storage.

To understand the principle behind pad clamps you have to understand how pads work. Your basic pad is a disc of felt ( about 5mm thick ) with a stiff base, usually card, enclosed in a skin - usually leather. To this is fitted a reflector, which helps to minimise loss of tone projection through absorption.
The pad's job is to make a seal over the tone hole. In order for it to be able to do this is has to be 'set'. This involves a number of procedures, the first of which is to check that the tone hole and pad cup is level. The pad is then fitted, adjustments are made to ensure the pad will lie flat against the tone hole ( either by selecting a pad of appropriate thickness, or adjusting that thickness - or by manipulating the angle of the key cup ).
Finally, the pad is heated, ironed and brought down onto the tone hole - which creates an indentation in the pad that corresponds to the tone hole. If this process has been done correctly the pad will seal evenly all the way round the hole.

There's a correlation between the pressure applied when setting the pad and its subsequent movement. The harder you set a pad, the deeper the seat..and the deeper the impression in the felt body. By overly compressing the felt you accentuate its potential to expand ( it's all in the fibres, dontcha know ). This is why so many brand new horns ( particularly student clarinets ) start leaking about three weeks after they've left the shop, as manufacturers these days tend to use compression techniques to set pads.
Once the pad has been set it is then at the mercy of environmental factors such as temperature and humidity...not to mention mechanical factors such as wear in the action.

To some degree pads can absorb the effects of these influences, by expanding or contracting to a greater or lesser degree.
On the face of it this sounds dreadful - but as long as the pad is evenly set ( and this is very important ) any such movement is evenly distributed. The generally held assumption is that pads, if left standing, will lose their 'seat'...they will no longer seal the tone hole. This is true, up to a point - if you leave a horn in storage for, say, a year or so, then there's a good chance that when you come to play it again you'll notice it perhaps blows a little stiffly ( this is why brand new horns are delivered to the shop with their keys wedged down with bits of 'shipping' cork ).
However, if the pads have been correctly set ( not overly compressed, and with flat cups and tone holes ) it's more than likely that they'll settle down to a good seal after a few days playing ( this is why a week after removing those shipping corks that the G# tends to go out of regulation ).
Pad clamps play on this assumption - and you can't fault the marketing by insisting that pads don't lose their seat if left standing ( though you'd better not mention the time factor involved ).

There's a far more insidious side to pad clamps though. If you've had a naff pad job done then sooner or later the pads are going to cry foul. As stated, a properly set pad can distribute any small amount of movement evenly - but a badly set pad, say one that bites more at the rear of the tone hole than at the front ( a very common error ) will be quite unable to accommodate even relatively small degrees of movement before a leak shows at the weak spot.
This can show in a matter of mere weeks, or less...and there surely can't be a single repairer who's never wrestled with a difficult pad all day, only to find that it's gone completely out the next ( oooh, those Cigar Cutter low Ds eh? ). But - if you pop a clamp onto the pad you negate the effects of any movement by applying a constant pressure on the pad against the tone hole. That's not to say that pad clamps stop pads from moving - oh no, they still absorb moisture etc. but the clamp forces the pad to move around the seal, as it were. So, from the point of view of seating, provided you've got a decent pad job to start with there should be no need for pad clamps at all.

There are other issues too. Consider that common problem - that of the sticky G# key. The G# key ( and the low Eb and C# ) sticks because, when idle, it remains in contact with the tone hole. No matter how fastidious you are with your cleaning/drying regimen there will always be a spot of moisture floating about in the bore of a recently played horn. Its this moisture that leads to sticky pads - and the pads that show the effect most are the ones that are held against the tone hole. It's frustrating enough having to cope with half a dozen sticky pads when picking up your horn...but by using pad clamps you effectively make every pad a closed pad in storage.

Some say that pad clamps increase the length of time a pad lasts. A pad is only as good as the skin that covers it ( assuming it seats in the first place ). There are a couple of factors that affect skin longevity - the most notable of which is that of moisture. Again, the pads that seem to go first are those that are held against the tone hole in storage...so it follows that pad clamps will advance this process of wear.
The boundary between the skin of a pad and the surface of a tone hole is not a pleasant place to be...a spot of moisture, some dollops of fat, a drop of acid, assorted bacteria...all typical components of saliva. This all acts together to deteriorate the skin covering the pad - so it make obvious sense to avoid prolonged contact with that boundary.

There's also the issue of regulation. Once a pad is set it effectively locks that key into a set amount of travel - from standing at rest to being fully closed. Many keys are linked together, and so it's obvious to see that they each rely on the other operating to a constant... ie, the distance a pad has to travel to completely close.
If you apply a constant pressure to the pad you may well increase the distance that pad has to travel before it fully closes. It'll barely be a millimetre or so - but by the time this is translated through the length of the key it will add up to a not insignificant amount of play at the rear of the key.
This leads to 'double action'. What this means in real life is that when, say, you press your F key down, it may not quite fully bring down the Aux.F key above it...or indeed it may be held off by the Aux.F key - due to a mismatch in the amount of travel each key requires. But surely if you clamp all the pads you increase the travel by the exact same distance? Well no, your low D pad has a far greater contact area than your Bis. Bb - therefore it distributes the pressure more widely...so your Bis. Bb pad settles in deeper.

Some might say that the clamps exert very little pressure at all - indeed, I had a horn in the workshop that the player had fitted with magnetic clamps ( they fit up the bore and pull the pads down by magnetic action on the reflectors...bit of a bummer if you have nickel or plastic reflectors! ). They exert little pressure on the pad - but then if there's so little pressure, what possible good can they be doing? That little pressure is certainly no match for any slight expansion caused by the absorption of water into the pad.
In other words, the pad clamps have to be reasonably strong to perform any function at all, and if they do perform that function then you have to account for the issues detailed above.

I've heard reports of people claiming that their horn worked much better after a set of clamps were applied. That's great, we all want working horns...but one has to wonder whether the horn was, in fact, working properly in the first place. So, personally I think they're a con - I certainly have never needed them...and either of my horns can go weeks without being played with no ill effects whatsoever. As for my repairs, I assume the client will not use them - I'd consider it something of a defeat if they felt they needed them after a repad - and consequently I regulate the instrument to the constant of a flat, well set pad.

If they have a use, as far as I'm concerned, then it's during periods of prolonged storage - or perhaps shipping.