The user manual for the (1980) Atari 2600 version of Space Invaders appears to be the first documented back-story for the aliens. In short, your task is to use your laser cannon to defend the Earth from an seemingly neverending alien fleet of invaders, bent on the invasion of Earth.
Each time you turn on SPACE INVADERS™ you will be at war with enemies
from space who are threatening the earth. Your objective is to destroy
these invaders by firing your "laser cannon." You must wipe out the
invaders either before they reach the earth (bottom of the screen), or
before they hit you three times with their "laser bombs."
Your long-term objective is to score as many points as possible.
Points are scored each time you hit one of the SPACE INVADERS. They
are worth different amounts of points, depending on their initial
position on the screen.
To answer your specific questions, the space invaders come
and while their motivation isn't spelled out, what's obvious from the subtext is that these invaders represent nothing less than an existential threat to the human race and must be stopped at all costs, even if that means your own death.