I think the reason we care is gossip. Gossip is part of being in a human community, it binds us, and polices our behavior to acknowledge norms. This works pretty well in smallish groups defined by nearly daily personal interaction. And important members of the tribe are defined in part by having a personal relationship with the most people and having the most people care about their lives. But modern mass communication hijacks this feature of human social behavior and helps us treat famous people as though they are part of our own tribe. We know things about their personal lives that we shouldn't given that they aren't actually in our social circle, but still we know these things and are drawn to learn more of these things because the other people in our tribe do too, as though the celebrities are important -- like tribal elders or great leaders or something -- even though they aren't any of those things in our actual social group/tribe.
Gossip can be destructive and mean. It can be used to intentionally spread lies or exaggerations about people to "put them in their place." And of course the policing of behavior can backfire on the excessive gossiper too, making them a source of scorn in the community. But even when used in means ways, the overall effect on the tribe is bonding, even if it is bonding in dislike of someone. I'm inclined to think of gossip less as a good or bad thing and more as mere part of what defines a social group, even playing a role in defining who is in the tribe and who isn't because no one knows or cares about them. We gossip about the people who matter to us and to our tribe -- but this sure does get weirdly applied in a mass communication culture of hundreds of millions of people. Hence celebrities we know a lot about but don't actually know and the painful obsession some people have in tearing celebrities down, of putting them in their place.
There is no doubt that there are famous people who are famous largely because of their own self promotion -- the Kardashians. But there are also plenty of performers who don't try to attract attention in their daily lives, but who are harassed endlessly by the press and the public. They can't go shopping, go on a date, take a vacation, etc. without ending up in the celebrity press -- their lives offered up to us to consume and gossip about, not because their daily lives are noteworthy, but because they are famous and therefore interesting to us. On the one hand, they are famous and possibly wealthy and certainly enviable -- giving up aspects of their privacy sure seems like a reasonable trade off. On the other hand, it must suck to be largely unable to do ordinary things without being harassed, to be critiqued by total strangers for every pound gained, for every bad haircut, or to have their romantic relationships scrutinized to a degree most of us wouldn't tolerate from our friends.
But, gossip isn't all bad. Sometimes it is used to spread information intended to help people or spare their feelings. Like if someone has suffered a personal tragedy that isn't well known. Gossip can spread that news without any need for the person who is suffering to have to keep telling people. And gossip can be used in that situation to encourage other members of the tribe to be nice to the person suffering. Still it is bonding of the tribe and its members, even though it is gossip. Just a feature of human interaction that can be good or bad depending on how it is used.