There's no consensus on whether burn-in is physical or merely psychological, though "psychological" deserves its share of respect. When you get a new pair of glasses, it can take time to adjust to the new prescription. That's not "psychological" so much as a processing issue with your brain. You can experience this if you A/B too many headphones in rapid succession. Your brain will actually get confused and everything will sound terrible.
On the other hand, many of us have noticed that our headphones sound better after a certain number of hours of usage. It's widely believed that running your headphones loosens up the diaphragm a little and actually improves the sound. There's no science behind this. No machine ever invented has been able to pick this up, but there are testimonies galore, for what it's worth.
Some people get very involved with burning in. They won't listen to their headphones until they've been "burned in" for hundreds of hours. Some people think you should crank the volume up to a high but tolerable level. Some people think you should use pink noise. A lot of people say you should just wear your headphones and enjoy them. Whatever the reason, it's common to wake up one day and say, "Wow, this sounds pretty good."
Whatever you do, be kind to your headphones. Running them at really high volumes, especially if there's some really hardcore bass, can damage the diaphragm. The best judge of volume is your set of ears. If you're wearing the headphones, you're not going to turn them louder than you can bare - even if you're crazy. But if you leave your headphones running, and you're not monitoring the volume, it's possible to push the drivers harder than advisable. You're more likely to "loosen" the driver up in a negative way, and end up with buzzing. It would be an extreme case where you actually punctured or ruptured the driver, but the best way to avoid that possibility is to use some common sense.
