"How do you do"--what does it mean and when did most of society opt to no longer say it?

"How do you do" is a simple greeting or salutation.

how do you do — a polite greeting that you can use when first introduced to someone

A more modern variant would be, "How are you?" or "Nice to meet you."

As for usage trends, NGrams can give us a small hint at when it began to phase out of style. Note the decline between 1900 and 1920:

It's an alternate construction of the question, "how are you?", or, "how fare you?"

Perhaps, "how do things go?". I also like, "how do you be?" (mentioned by d'alar'cop).

Akin to the German "Wie geht es Ihnen?" (semantically, if not in terms of formality).

At school in Britain in the 1950s we were taught not to say 'nice to meet you', or worse still, when someone said 'How do you do', NEVER to reply 'nicely thank you'. They were supposed to have been bad form. We were taught to say a polite 'How do you do' in return.

And that is what I still say today! And so do many other people. In other words I am not sure that people have stopped saying it.