That's not an idiom. It's not even an expression. I assume you have seen it in a full sentence. It would be helpful if you posted the whole sentence.
On the assumption that it said something like "He could not bear to hear the news" then it means that he couldn't stand to hear the news, he couldn't endure it, he had no tolerance for it. It is simply a combination of the verbs "to bear" and "to hear" in one sentence.