Others have discussed strength training/conditioning/physiology, and remediation techniques like cool water baths. There are two aggrevating factors that haven't been addressed:
If you use good technique then you will support the maximum amount of your weight with your leg muscles. The obvious benefit of supporting more weight with your legs is your arm muscles don't have to work so hard. As we all know the less work you do the more you can do it. Another technique to give you more stamina is to favor positions where your bone structure is supporting your weight instead of your muscles. Finally avoid odd angles when gripping holds. An odd angle with your elbow pointing up at the sky will require more muscle tension in order to sustain, and require more energy/effort. Other technical mistakes also come into play here but these are the ones I've fallen victim to most often.
When people are nervous they then to grip holds like their life depends on it. Consequently they'll apply 2x the force needed to maintain their grip on the hold. To put it another way when you're talking on your cellphone I can sneak up behind you and easily snatch it out of your hand because you're only using the amount of force required to prevent the phone from falling out of your hands. You aren't using the amount of force required to prevent me from snatching it out of your hands. Imaging how tiring it would be to maintain a death grip on your cellphone for a 20 minute conversation. Now apply that to climbing. I'm willing to bet that most climbers can afford to "loosen up a bit" when they're out on the rocks.
Of course there are (not so) positive psychological feedback loops at play here which reduce your effectiveness. People are nervous about their ability to do XYZ on the rock so they grip the holds a little harder, which makes them feel more tired, which makes them want to hurry to complete the moves, which makes them lose focus on their technique, which makes them try to power through moves, which makes them more tired, which makes them nervous, and the cycle repeats. That is why the mental aspect of climbing is crucial, especially at the higher levels where there is less margin for error.