To rekindle a feeling is to make it come alive again after it has gone quiet or cold. People rekindle a romance, an old friendship, interest in a subject, hope, or even anger. The picture is a fire: kindle means to start a fire; rekindle means to get the flames going again. You will hear this a lot in magazines and films about love, but it works for any feeling or thought that had faded. Though actually, you can still use it of a real fire, but that use is less common now. The word is not slang, and it is not very formal either — it is the ordinary, slightly warm way to talk about something coming back.