He stopped doing something. I assume from the context that he stopped giving blood. He only had one reason for stopping. He only stopped (giving blood) because he became too old to do so. He wanted to continue but people his age are not allowed to give blood.

I'm not sure I understand your question. The sentence is correct as is. "only stopped because . . X" means "X was the only reason he stopped". Maybe the word order is confusing. You could say: He stopped only because . . . OR He only stopped because . . . They're the same.