R.F.D. stands for Rural Free Delivery. In the years preceding and during the time The Andy Griffith Show was on the post office delivered mail to rural areas without extra cost to the reciever.
Mayberry is the name of the town it took place in. RFD stands for Rural Free Delivery, which was a new program of the federal government where postmen would bring letters to your house instead of you having to come to the post office to get them. At one time, in the USA, only urban areas had home delivery of mail and rural residents had to come to the Post Office to get their mail.
The name Mayberry RFD is like Beverly Hills 90210, then.