When USPS mail says return service requested such as on a bill etc.
The USPS return service requested on the mail means that the sender is requesting return service and wants the USPS to return the mail back to them in the event the mail is not deliverable to the addressee.
The USPS then informs the sender of the mail on why the return service requested mail was not deliverable.
Return Service Requested on USPS mail such as bills, letters etc is used by USPS in the event that the mail is undeliverable or forwarded and so that a notice is mailed to the sender with information how it was undeliverable - usually for lower than first class mail.
So if the mail is sent to someone and it's not able to be forwarded or delivered to the addressee then it's sent back with information on why it was not able to be delivered.
If you move you need to let places that send you mail know of your new address even though you have your mail forwarded.
It's still your responsibility to notify the senders of the mail of your correct address so that the mail gets to your address on time and correctly.