In Aliens vs Predator, and the other Alien movies, how did the aliens get off their home planet?
Assuming I'm correctly understanding your question, it was established in the film that the Xenomorph eggs were brought by the Predators. It is unknown whether the Predators directly harvested said eggs, or perhaps acquired them from another race.
Through translation of the pyramid's hieroglyphs, Alexa and Sebastian learn that the Predators have been visiting Earth for thousands of years. It was they who taught early human civilizations how to build pyramids, and were worshiped as gods. Every 100 years they would visit Earth to take part in a rite of passage in which several humans would sacrifice themselves as hosts for the Aliens, creating the "ultimate prey" for the Predators to hunt. If overwhelmed, the Predators would activate a self-destruct device to eliminate the Aliens and themselves.
@FuzzyBoots answers pretty nicely sums it up, but let me add few more words:
The Alien movie shows a space ship operated by Space Jockey (probably the "Engineers" from "Prometheus") that contained alien eggs, similar looking structure is visible in the first "Alien vs Predator" game (there is also a giant statue of Xenomorph, which started the rumors that indeed there was deeper connection between Engineers and Aliens).
Beside the "AvP" movies, comic books describe the Predators capturing queen and harvesting her eggs for their hunts in more detail and later they spread it over planets that have a lifeforms able to participate in xenomorph life cycle (its worth to point that predators were trying to be careful to not use the queen egg).I remember also panel in one of the comic displaying another alien creature stealing alien egg and later succumbing to the facehugger.
Aliens by themselves are absolutely not able to use tools, not even mentioning the space travel. Their intelligence level is changing depending on the author - usually they are described barely as a smart animals, but in the Aliens - labyrinth dr Paul Church says, that he was a prisoner in dying alien hive for long period of time and says that xenomorphs were intelligent enough to attempt to develop the cure for the devastating disease and find a mean to keep and breed humans as cattle (but then, many things point that he is a very unreliable narrator).
The principal theory of their connection to the Xenomorphs, which was mentioned briefly by Ridley Scott in his director's commentary for the first Alien DVD, is that the Engineer's Ship in Alien was a "bomber" and that they used them as biogenic weapons to fight an ancient war against an unknown foe. There is some evidence to support this, such as the Xenomorph's biomechanical nature. Xenomorph Eggs were believed to be used as "bombs" on an enemy planet and then the Xenomorphs would proceed to kill the entire population as they spawned.
To answer you question, Engineers have ships, so the scattering of the Alien species wasn't that hard.
In my opinion, the Xenomorph homeworld Xenomorph Prime is the planet the Engineers used for "storage", given that the planet has other inhabinant species.