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.