How did Goku become a super saiyan

he became a super saiyan by getting enough rage i nthe freiza saga when freiza exploding krillen into chunky bits of flesh all of the anger unlocked his legendary warrior spirit.

To be more specific, his body as simply put was ready for the transformation. At that point Goku was far and beyond the full limits of a pure blooded saiyan, and Goku himself thought that he had also trained to his bodies limits and has gotten as strong as any man or saiyan could, he technically was right he'd reached his limit. As we all know saiyans were subject to large bursts of power when angered, once frieza killed krillin goku's anger burst open and past the limits of his body. Basically the transformation pushed his limit to a whole new level past his original.