A fan, in this context is a great admirer of a person or group of people (such as an actor, musician or band.) It comes from the word fanatic, which means an over-enthusiastic believer in a religion or other belief system.
The verb to rave means to talk wildly or furiously, or to talk with extreme excitement and enthusiasm about something. In this phrase, the latter meaning is the right one. "Raving" also has the meaning of being insane, out of control, or completely over the top. "Raving mad" means totally insane, and a "raving beauty" is someone so beautiful that it sends people crazy. If you are a raving fan of (for example) a singer, you are absolutely crazy about him or her -so much that you can hardly control yourself. Of course, people are often joking a bit when they use this phrase.
The verb to rave means to talk wildly or furiously, or to talk with extreme excitement and enthusiasm about something. In this phrase, the latter meaning is the right one. "Raving" also has the meaning of being insane, out of control, or completely over the top. "Raving mad" means totally insane, and a "raving beauty" is someone so beautiful that it sends people crazy. If you are a raving fan of (for example) a singer, you are absolutely crazy about him or her -so much that you can hardly control yourself. Of course, people are often joking a bit when they use this phrase.