It has always been the policy of the US postal service to require a return address. The first letter I remember mailing as a little boy almost 40 years ago was rejected for the same reason.
The return address does not have to be your address. You can make it the actual destination, if you wish. The reason for the return address is to help prevent mail from ending up in the dead letter section. Even though the address is correct, it could get smeared, smudged, or ripped off the envelope entirely en route. Without the return address the post office would not know where to return your mail. Also, the intended recipient could move without a forwarding address.
You can make the return address the local post office, if you live in a sufficiently small community, and do not want a relative to know your address. Or you could make it the address of another relative you trust not to divulge YOUR address should the letter prove undeliverable.
I have mailed envelopes to a post office that contained a letter with the actual address. That way the person you're sending mail to won't even have a post mark from the city you are sending from. This, however, costs 82 cents instead of 41 cents (obviously).
Finally, you could also send your letter Federal Express for a mere $15. I think they prefer a return address also, however.
Addendum: You might think Pegasus is kidding, but the US government runs foreign gulags with torture policies far more excessive than the mild water boarding and blanket parties practiced at Guantanamo Bay. Furthermore, the President has authorized himself to read your mail without a warrant. You could be declared an "enemy combatant" and extremely rendited without recourse to habeaus corpus, held incommunicado, tortured at some foreign gulag, and all your assets seized, based solely upon the president's say so. So far we know of only a very small handful of american citizens to whom this has happened, but then, the actual number is also secret.
Joseph Stalin's favorite expression was "no man, no problem." So far this president has not killed anywhere near as many people as Joseph Stalin, but he has granted the power to behave with relative impugnity to whomever the next president might be. We can all hope and pray the american electorate makes a much better decision next time.