How do I connect the electrical ribbon to this metal contact?

Use X-Zacto knife to scrape off outer plastic on ribbon cable to expose copper lead.

Next you will need a soldering iron,solder and a flux pen. Clean places to be soldered together with alcohol then apply flux from flux pen to them. Even though solder has flux in it this step is needed. Apply some solder to end of each lead on ribbon cable. Lay ribbon cable on top of corresponding place on board. Then touch with soldering iron just long enough to make connection.

I surely think that the ribbon cable would plug into a device on the other side of the circuit board.

Usually you press down on the device and place the ribbon cable in and release the device and it is locked in.

You can't. Ideally, you should replace it with a similar ribbon cable with the terminals intact. If not, you can try using 4 very thin wires to replace the whole ribbon altogether.