Начал было -- nearly started. It means he started to tell, but suddenly stopped.
Начал бы -- would started. He would started to tell, but suddenly stopped. This senseless is meaningless because he is not WOULD started to tell, he already started to tell.
We say would (бы) when think what could happen if he did not hat he did in the past, if instead if it he did something another (for example: "Did you come to me? It would be better you to kill yourself.)"
In our case, He started to tell - it's the fact, which already happened. That is why we don;t use начал бы. Using method of exception we choose начал было. In this sentence начал было has the same meaning as начал бы.