Why and how has fame replaced worth as our society's yardstick when judging people?
First of all, I think that it is highly speculative to say that fame has replaced worth as our basis for judging people. The question implies that we are shallower than people in past times and that they judged people more accurately than we do today. I think that it would be exceedingly hard to prove scientifically that this is indeed true and I am inclined to doubt that it is.
If we have to argue that fame has indeed overcome worth, I would say that we could blame it on the profusion of media that we have today. That, combined with our basic human narcissism, makes fame seem more appealing and easier to achieve than worth. In today’s world, we have many ways of becoming famous. These have been provided to us by television (such as by reality TV) and perhaps even more by the internet. Anyone can hope to become the star of a viral video sensation. Because we all want to be recognized and important, we seek fame.
Because we are able to seek fame, and because fame brings recognition with it, we come to value it more than we value worth. Worth is harder to judge and it is therefore less likely to be acknowledged by society. This is what we can argue if we have to argue that fame is more important than worth.