Using 'Signed' as a Closing

It's used by the person reading the letter out loud in order to make it clear that the name is formatted on the letter as a signature.

And, as jsw29 states,'No, signed is not normally used in writing one's own letters in the way proposed by the OP'

It is not listed here as a common "complimentary close" . I suspect it is only used if you are reading someone else's letter out loud.

As Hot Licks's answer states, what you have observed is just something done when reading aloud to indicate a signature.

However, you may sometimes encounter text like '[signed on original]' in a written document being distributed electronically, to indicate that the document has been signed on some original or official copy but that the electronic copy does not have a signature. I have encountered this frequently in British military documents, for example daily orders which would normally have been signed and then photocopied but are now more likely to be emailed.

While other answers address your question, there is a curious and somewhat related fact: sometimes in medical letters you may find the closing words "not signed". An example I saw only yesterday closes with:

Best wishes,
Dr Surname GP
Electronically checked but not signed

This indicates that the message was dictated to a secretary, or otherwise was not written personally by the doctor, and it has not been read by the doctor (otherwise they would have signed it).