It's an act of trust and a way to show you they value you/how much you mean to them. Even if the emotions expressed are negative. They're not trying to blame or expect you to make everything better magically. Maybe just acknowledge their feelings or show that you understand? Just talk with them, they want to explore the feeling with you (thoroughly), they don't necessarily expect a fix or an endpoint. Like Uumlau said, there really is no task at hand. It's like Ti vs. Te.But then if they are not reciprocating but instead just keep expressing to you their feelings without engaging with constructive suggestions, then they are just being self-indulgent and abusing my sense of obligation.
Of course there's a point at which regardless of the kind of feeling you use, you start to indulge in whining. A good Fi user will realise when they start to burden you with their emo spewing or at least empathise with your position.