You hold the puzzle so that it looks like a 3 ring version of the Olympic rings. Both ends of the cord are attached to rings hanging below the rack, and those rings should be on the backside of the rack. The right cord ring and left cord ring are attached to the right cord and left cord respectively. The ring that is being taken off is in the center, trapped by the 2 cords that go up, over, and back down to the rings on either side. There are left-handed and right-handed versions that depend upon the orientation of the cord coming over the notch in the rack. If the right cord is the one that comes up from the back, over and down through the 2 rings on the left, then it is a left-handed version, which is what is described here. For right-handed versions, swap left and right as described below.
Hold the puzzle horizontally with the rack pointing away from you. Hold the center ring against the right ring with your right thumb. Grab the cord at the notch where it comes over the rack and bring it up to the left and over the left rack post allowing the right cord to pass under the rack. Pull the cord left forming a loop with the right cord under the puzzle. Swing the loop down and forward with the right cord passing under the left cord ring. This causes the right cord to come forward, and it should slip out from under the left ring. Now both cords lie on top of the left ring. Still holding the loop, push it down through the left ring. The left cord ring can now be brought forward through the loop and deposited in front of the right cord. Let go of the loop, pull the left cord ring, and the center ring will untangle and come off.
Reverse directions to get it back on. The cord is now draped between the two rings attached to the rack. Place the loose ring over the cord and lift the cord through the ring. Twist it counterclockwise slightly to keep the left cord in front and the right cord in back, to form a loop. Push the loop down through the left ring. Pass the left cord ring through the loop and deposit it behind the right cord. Flip the center ring up and untangle cords. The right cord is now clearly in front and the left cord is in back. Pull the loop up out of the left ring. Slip the right cord under the left ring keeping the loop on the left side. Pass the right cord under the left cord ring and bring the loop all the way up on the left side, over the top of the rack, and deposit it in the notch in the center. Flip the trapped ring down to form the Olympic rings pattern. The puzzle has been restored.