I think it's a fair question to ask, and a good introductory one. And I'm sure we all have different reasons.
Perhaps you saw one of her films on TV or on video and were hooked, either by her looks or her acting (or both). Maybe you read about her and were fascinated. What is it about her that seems special, and makes you a fan of someone whose death likely occurred well before you were born?
For me, it's her timelessness -- not just her beauty, but her attitude. Carole's iconoclastic nature resonates several generations after she walked this earth. She exuded sex appeal, but in a feminist fashion; the lady was athletic and competitive, qualities that speak to us today. But she was also down-to-earth, generous, caring. She may have been one of the giants of Hollywood, but she never acted like one, which was part of her charm. Lombard is the type of person we say we would be if we somehow became a celebrity. She clearly enjoyed her fame, but invariably used its benefits in a positive manner. Her acting also holds up well these many decades later, as she remains the standard by whom all comedic actresses are measured.
So, let's have your comments on this. Just why are you a Carole Lombard fan? As the saying goes, inquiring minds want to know.