Great question — answering as the UphillTech team:
Your bike has V Brakes, so the two arms should swing in as you squeeze the lever. They should pull back off the rim under spring tension. Check the wire moves in the housing (rust). The shifter has at least two runs of exposed inner wiring (silver, though likely rusty by now) Try and pull those bits with your fingers and see if the rear derailleur moves. I suspect the outer housing has failed and the shifter can't pull enough cable to make the RD move. A photo of your bike would be useful as a comparison. Good luck !
Full answer by Criggie: https://bicycles.stackexchange.com/a/97326 (CC BY-SA 4.0).
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