The right answer is "начал было", because "начал бы" is the main clause of the conditional sentence. It means "He would have began..." (he would have begun to do something under a certain condition but he didn't.). The right answer is "начал было" which means "He began to tell something, but suddenly stopped."

"Начал было" is properly translated as "nearly started". "он начал было рассказывать,но вдруг замочал" "he had nearly started telling it but suddenly lapsed into silence" "Начал бы" is really a part of the conditional. Consider the following: "он начал бы рассказывать, если бы его не прервали" "he would have started telling it if he was not interrupted"

он начал было рассказывать, но замоЛчал) (вы пропустили букву Л в слове)

Он начал было рассказывать, но вдруг замолчал. Он только хотел рассказать, но вдруг замолк.

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.