One of the most controversial (and obscure) aspects of Goku is so powerful, but this explanation, provided by Akira Toriyama himself, is still one of the most hated retcons in the history of Dragon Ball.
The original Super Saiyan transformation happened during when Vegeta then transformed too, the whole 'pure heart' requirement was put into question. Later in the series, the young sons of Goku and Vegeta, Goten and Trunks, were able to transform while still kids and without experiencing any significant emotional turmoil, which made the initial Super Saiyan explanation even less believable.
Many years later, when the sequel Dragon Ball Super was in full swing, series creator special cells called S-Cells, which, if in high enough concentration, can trigger the transformation into Super Saiyan. Any strong emotion can activate S-Cells, not just rage. The amount of these cells is higher in Saiyans who have a gentle attitude (hence the 'pure heart' idea) and high battle power, which is why Goku was the perfect candidate, while Vegeta likely inherited a great quantity of S-Cells from his 'pure' bloodline. Trunks and Goten inherited the high concentration of S-Cells from their fathers, and they also grew up on the peaceful Earth developing kind personalities, which explains how they were able to transform at such a young age.
S-Cells Are A Controversial Retcon, But They Still Make Sense
Toriyama provided this explanation to the appearance of Trunks and Goten's early transformations. The author actually explained these at the time by saying that human-Saiyan hybrids are particularly powerful, which makes sense in retrospect, considering that Earthlings are peaceful and kind, compared to Saiyans, so the kids inherited these traits.
Some fans also pointed out that the S-Cells theory bears a striking resemblance with the Midi-chlorians, another very famous retcon, from the Star Wars franchise, that is not very appreciated by fans. It is indeed a bit disappointing to have Goku's iconic transformation come from a genetic trait rather than from a tragic experience and awakening. However, Toriyama made sure to add that a gentle spirit alone is not enough to increase the number of S-Cells, one also needs the battle power that comes from continuous fighting and training. So, while the S-Cells retcon remains one of the most controversial in Dragon Ball history, this should not be considered the only reason for Goku's unique strength.
Source: Akira Toriyama interview from Saikyō Jump, January 2018 issue, retrieved from Kanzenshuu.com.
