Most Saiyan (Goku, Vegeta, Gohan, etc.) who evolved into Super Saiyan had trouble fulfilling one of the requirements to evolve: Their rage or other emotions had to be extreme.
However, there are two exceptions: Goten and Trunks (not the Trunks from the future). How did they achieve it? They hadn't seen or done anything that could drive them to such a level of rage/intense emotion to allow them to turn into Super Saiyan.
According to this Yahoo answer:
Because by the time they were born Goku and Vegeta had already been Super Saiyan and with the higher power level thus when trunks and goten where born their levels were also higher.
Think about it this way Vegeta (Fut Trunks timeline) was weaker at SSj Level so Future Trunks took longer to become ssj (i think he was 13) while Vegeta (regular DBz timeline) had trained harder by knowing the androids arrival 3 yrs prior to chibi trunks being born thus more powerful ssj1 vegeta. And Goten being born around some time near end of the Cell Games saga Goku had became already full power ssj.
Thus as children their power levels were already exponentially higher than the latter of gohan who was born back when goku was weaker in power level.
Basically Chi-Chi gave birth to a Demigod.
Genetically speaking, Saiyans aren't human. So it's hard to say that by becoming Super Saiyans, Goku and Vegeta have altered the DNA in their sperm to be able to pass along those traits. I think the traits have been their all along, or maybe they are just traits that lay dormant in every Saiyan ultimately but most just never realize the potential. Even though supposedly legend spoke of their being only one true Super Saiyan, every culture has it's lore.
That being said, children are definitely more susceptible to believing anything is possible. Goten and Trunks are brought into a universe different from the one their fathers grew up in. For them, Super Saiyans are no longer these "mythical beings" legends had everyone believe. Psychologically, achieving Super Sayian status was completely plausible in their young minds, and mentality did seem to carry a lot of weight in pushing Sayians to that super status. While it could be caused by anger or any other strong powerful emotion, for them it was just youthful passion for strength and an absence of doubt.
This has been directly addressed by Akira Toriyama in an interview. Links to the translated interview can be found in this Kotaku article.
He addresses the case of Goten and Trunks here:
It's not like anyone can become a Super Saiyan through training and anger. In order to become a Super Saiyan, one's body must contain something called "S-Cells". Once these S-Cells reach a certain amount, a trigger such as anger will explosively increase the S-Cells and cause a change in the body: that's Super Saiyan. Most Saiyans have some S-Cells, although not a great quantity. The reason why Goku and Vegeta's children can become Super Saiyan relatively easily is probably because to a certain extent they inherited a lot of S-Cells, and also because Earth's environment is gentler and easier to live in than Planet Vegeta.
He goes on to say:
Having a gentle spirit is the best way to increase one's S-Cells, but most Saiyans have trouble with this, which I think is why no Super Saiyans appeared for such a long time and they became the stuff of legend. However, one can't reach the quantity necessary for becoming a Super Saiyan simply by having a gentle spirit, so a certain amount of battle power is indeed necessary. Looked at in this light, it's easy to see why becoming a Super Saiyan came easy for Goku.
So, it seems that having a gentle spirit along with a high power level increases the "S-Cells" in a Saiyan's body, and allows their children to become Super Saiyan more easily. This also explains why Goten was able to become a Super Saiyan even more easily than Gohan - when Goten was conceived, Goku's power level was far higher and he had already become a Super Saiyan. In the case of Gohan, Goku was far weaker at the time of conception.
There doesn't seem to be an official reference, but I suppose there are three possible explanations (well, except the "the author wanted it this way" one):
Heritage: When they had them, their fathers already had managed to reach the SSJ stage. So they sort of passed the ability on and made it easier. This is true for Goku > Gohan > Goten and also for Vegeta > Trunks. This might mean that the base power level of their sons is higher than their.
Generations: The more you go on, the stronger they are. Gohan is stronger than Goku and Vegeta. He gets weak only because he doesn't train enough. Goten and Trunks are even stronger.
Blood purity: It is unclear, but it seems that hybrids are stronger than pure-blood Saiyans. Apparently, hybrids lack the will of combat these have, though. And indeed, Gohan doesn't train as much in peace time.
A different view from what was already posted: because Saiyans get stronger the stronger their opponents are.
Goku, Vegeta and Gohan didn't fight Super Saiyan level opponents in their childhood. Gohan became a Super Saiyan while training with his father in the time chamber, and both Goten and Trunks trained with Gohan and Vegeta, who were already on Super Saiyan level at the time, so that could have made the kids gain more experience than their parents could.
They were born with there parents blood and got to train with there parents Trunks with Vegeta with the gravity ship and Goten with Gohan training together while he was also learning how to fly.
