When you "try to talk some sense into someone", you are trying to convince them that a choice they are making or a direction in which they are taking their life is wrong and should be changed.
In the above sentence, the speaker thinks the the woman is making a big mistake by break up with "him" - a third person in this case. We assume "him" refers to a boyfriend or husband she is about to break up with.
You usually use this phrase when you think that what the person is doing is very obviously wrong.
"He wanted to buy that motorcycle when he has never ridden one. I've tried to talk some sense to him, but it didn't help."