How did Goku and Vegeta alone end up on Earth? 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?
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


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