How does my brain know to wake up just before my alarm goes off, especially on the morning of something important.

Your body has internal cues in the form of circadian rhythms (including your sleep-wake cycle, your body temperature, hormone levels, and impulses from nerves in your stomach, etc) which all feed back up to the supra chiasmic nucleus in your brain which is a bit like the clock face - get rid of it and you can't tell the time) that help you wake up. But these are also often attuned to environmental cues - light, sound (e.g. birds singing), temperature, etc.

Chances are you have conditioned yourself to wake up in relation to some of these cues which you may be more aware of as you exit REM sleep and go back into a combination of stage 1 sleep / wakefulness. As such, your mind may become aware of birds singing, light coming into your room, body temperature, changes in adrenal and noradrenal hormones, having had a certain amount of REM sleep cycles or stomach emptiness, all of which are cues (then there's the often forgotten glimpses of the alarm clock while we are half asleep!).