Women are advised to try to physically relax during labor, while the cervix is still dilating. That is, to "breath away" the contractions - the puffing thing. Why? Pain is (in my personal experience) easier dealth with with clenched teeth. Actually, i only survived the worst part of giving birth to my son, with my back arched and my hamstrings tightened to the max, on each contraction.
While I think this is a very complicated subject with no one good answer - both because each woman will have a different experience and each woman will benefit from different guidance - there is one answer to the question.
The "relax" instruction is not in relation to pain management specifically, but rather in order to help allow the cervix to dilate. Being too "tight" can cause a delay in dilation - as can being too relaxed (the latter seemed to be the case for my wife in her first delivery, where she spent a few hours in the shower and that seemed to stop the dilation).
This is supported in several online resources, such as Babycentre.co.uk. They list that while stressed, the following occur:
It's natural to feel a bit anxious. But if you feel very stressed it can interfere with your labour. Stress and anxiety make your body produce fight-or-flight hormones, such as adrenaline. These stress hormones may:
- Reduce blood flow to your uterus (womb).
- Suppress the release of oxytocin.
- Slow your first stage of labour down.
Pain management is a separate issue, although obviously also very important, and to some degree will help reduce your stress on its own - so if clenching your teeth helps you deal with pain, I say go for it. Much of this is really intended for the first stage and early second stage, in any event; by the time you're in transition, I doubt there's much you can do to truly relax, but you're also hopefully progressing smoothly at that point.
Child labor is already a physically-intensive activity.
Intensely clenching muscles furthers the body's exhaustion. Since hamstring, quadricep, and back muscles are some of the largest muscles in the body, use of them can be more exhausting. Exhaustion can increase irritability, agitation, discomfort, among other undesirable effects.
Furthermore, strongly clenching your muscles in this physically agitated state may increase your chance of skeletomuscular injury, such as pulling a muscle are hurting your back. If you happen to have an epidural, back-clenching could cause other complications with the insertion site.
Using your muscles this way, or clenching them, can potentially affect your respiration, heart rate and blood pressure adversely (considering your body is already attempting to deliver). Having controlled oxygen levels and bloodpressure is important both for the mother's health and the baby's health during delivery. Unfortunately, I don't have specific sources available for this conjecture, because there aren't studies on muscle contraction specifically during labor. However, sustained muscle contraction like flexing your legs is essentially isometric exercise, and you can make assumptions from that point.
There are also pain theories out there, such as the pain-fear-tension triangle theory, that would advise tension increases pain which increases fear which increases tension and so on. While this appears not to apply to you, it appears to be a very popular theory (at least in western medicine and the midwife/homebirth type of communities). The physiology behind this would be that your contraction of muscles in or around abdomen impair the ability of the uterine muscles to contract. This, then, would impair the labor process and/or increase pain.
Other theories suggest the tension has an affect on different hormone levels. For instance, if your stress hormones are increasing then they are directly impairing the ability of the necessary oxytocin hormone. Joe's answer covers this well.
I believe another reason they ask mothers to relax is to make them easier to manage. Not just so they can tell the mother what to do, but so the mother is in a state where she can more clearly and accurately assess and communicate her needs. It may also help the medical/delivery professionals differentiate between stress-related symptoms and symptoms indicative of medical complications.
Note: Not a doctor, so this isn't medical advice.
Consider what's basically going on; your body is attempting to build up a rhythm that will cause your child to exit in a calm and controlled fashion. The ideal process would be muscles smoothly grasping and shoving the baby down and out in one smooth motion.
Obviously the issue here is that, because the natural process of childbirth includes squeezing a 10-pound baby through a channel barely adequate for the process that's never been exercised this way before, it hurts like hell, so the other natural response is to tense up, in order to prevent any changes to the current "manageable" state of your body.
Unfortunately for all involved, you don't want to prevent changes. You want the situation to change from "baby in transit" to "baby looking out at the world wondering what the dickens just happened". Tensing the muscles that are pushing the baby out utterly screws up the whole "grasp and push" plan. Gritting your teeth and thus stopping pulling oxygen in causes your muscles to run out of oxygen and stop doing what they were meant to do.
So you've got two competing natural instincts; "stop the muscles moving" and "smooth flexing of the muscles". The doctor's advice is to attempt to support the second, because that's the only way this process is going to end. Also, if you can do it smoothly, the pain will be less, so the muscles will tense less, so there'll be less pain and the whole thing will go more smoothly.
So the advice is basically sound. Obviously, this would be easier done if you got to practice beforehand, weren't in a completely new situation, and had any respect for the idiots around you trying to explain what to do while you are ACTUALLY IN SOME SERIOUS PAIN RIGHT NOW GO AWAY THANK YOU. This is why other advice includes Kegel Exercises and other methods of training your baby shoving muscles to cope with today's ridiculous experience.