Why Do You Wake Up Before Your Alarm Clock Goes Off?

Because your brain wants to wake you up on its schedule:

At the center of your brain, a clump of nerves—called the suprachiasmatic nucleus—oversees your body’s clock: the circadian rhythm. It determines when you feel sleepy and when you feel bright-eyed. It controls your blood pressure, your body temperature, and your sense of time. It turns your body into a finely tuned machine.
That machine happens to love predictability. Your body is most efficient when there’s a routine to follow. So if you hit the hay the same time each night and awake the same time each morning, your body locks that behavior in. And that’s where things get sciency.

Your sleep-wake cycle is regulated by a protein called PER. The protein level rises and falls each day, peaking in the evening and plummeting at night. When PER levels are low, your blood pressure drops, heart rate slows, and thinking becomes foggier. You get sleepy. If you follow a diligent sleep routine—waking up the same time every day—your body learns to increase your PER levels in time for your alarm. About an hour before you’re supposed to wake up, PER levels rise (along with your body temperature and blood pressure). To prepare for the stress of waking, your body releases a cocktail of stress hormones, like cortisol. Gradually, your sleep becomes lighter and lighter.
And that’s why you wake up before your alarm. Your body hates your alarm clock. It’s jarring. It’s stressful. And it ruins all that hard work. It defeats the purpose of gradually waking up. So, to avoid being interrupted, your body does something amazing: It starts increasing PER and stress hormones earlier in the night. Your body gets a head start so the waking process isn’t cut short. It’s so precise that your eyelids open minutes—maybe even seconds—before the alarm goes off.
There’s evidence you can will yourself to wake on time, too. Sleep scientists at Germany’s University of Lubeck asked 15 volunteers to sleep in their lab for three nights. One night, the group was told they’d be woken at 6 a.m., while on other nights the group was told they’d be woken at 9 a.m..
But the researchers lied—they woke the volunteers at 6 a.m anyway. And the results were startling. The days when sleepers were told they’d wake up early, their stress hormones increased at 4:30 a.m., as if they were anticipating an early morning. When the sleepers were told they’d wake up at 9 a.m., their stress hormones didn’t increase—and they woke up groggier.

I definitely believe the part aboutsometimes waking up when you know what time you need to be up. I've pretty much never slept until my alarm goes off  on a day when I'm leaving for vacation, even if I'm getting up a couple hours ahead of normal in order to catch a very early flight.  That always seemed weird to me, but I guess I understand why it happens, now.