Mirror lockup is advised in long exposure to avoid the vibration at the beginning of the exposure. This is what happens when you press the shutter:
1- Mirror goes up
2- Shutter opens
3- Sensor/Film is exposed
4- Shutter closes
5- Mirror goes down
As you can see from the sequence 4 & 5 the mirror lockup won't have any effect after the shutter closes and ends the exposure
But from sequence 1, 2, and 3 if the mirror goes up action cause vibration in the camera body and the shutter is getting open, the vibration will translate and captured in the exposure.
Now back to main question is why it's crucial in long exposure? Because if your shutter speed is 1/1000 of a second, it's faster than the vibration, the shutter will open and close before any vibration or camera shake could change the exposure. But if your exposure is 10 seconds, and the vibration cause by mirror lock up lasts for a second, then this shake will go to the first second of the exposure.
if short shutter speed is not faster than the vibration wave, it may be only one cycle of it which is okay and will look like there's no vibration happened.
A Wikipedia page that may help