How did Goku and Vegeta alone end up on Earth?

Question

How is it that only Goku and Vegeta ended up on Earth? What happened to the other Saiyans?

Planet Vegeta was destroyed by Frieza, but which planet did Goku come from?

Answer 1

At the same time Planet Plant (AKA Planet Vegeta) was being destroyed, Goku was being sent to a planet called Earth in order to kill all of it's inhabitants and then sell it to Frieza. Vegeta and Nappa were on another planet and Frieza told them that a huge meteor fell on their planet and destroyed it. That's how Goku and Vegeta survived. So that's make the survivors

• Goku
• Vegeta
• Nappa
• Raditz, Goku's older brother. It wasn't revealed how he survived.

And from the movies:

• Broly
• Paragus
• Turles
• Tarble, Vegeta's younger brother.

For more information read this page on Dragon Ball Wiki: Genicode of the Saiyans

How they ended up in Earth

1. Raditz goes to Earth to make Goku join them.
2. Goku disagrees but Raditz enforces him by taking his son, Gohan as a hostage
3. Goku and Piccolo kill Raditz
4. Vegeta hear about the dragon balls
5. Vegeta goes to Earth to find the Dragon Balls and then to destroy the Earth, he ends up fighting Goku

Answer 2

Planet Vegeta was destroyed by Frieza, yes. Goku and Vegeta are from the same planet, Goku was sent to earth as a child to destroy it. When Planet Vegeta was destroyed, Vegeta wasn't on the planet, he was away on an assignment at the time and Vegeta comes to Earth when he receives Raditz's dying report of seven magic objects known as the Dragon Balls which will grant wishes.

Other saiyans who made it to earth were Nappa and Raditz who were both killed.