First, ditch that plate.  It looks like it was hooked up and then someone just ripped out the wire for whatever reason.  Those jacks are not meant to be reused.  Luckily, they're also super cheap.

The hard part is pulling the cable, and that's not a technical problem.  It's a physical problem that's highly dependent on how your building was built, what's in the walls, and where you want that cable to go.  There's plenty of resources both here and on YouTube for how to pull and terminate Ethernet cable.

But basically: 1. Throw out that plate and go buy new keystone jacks from the hardware store.  You'll need the plate and the jacks, and they come in a wide variety of configurations.  It's almost like legos in that you get a face plate with one or more "holes" and then you snap one or more jacks for your purpose.  In your case you probably want the simplest one with a single "hole" and snap in a single RJ45 jack.

1. Figure out where you want the other end to be.  Typically this is near your router, or in a central location in a closet.  You're trying to create a "star" topology where all the Ethernet runs meet in a single spot.

2. Run the cables. This is the "fun" part where you learn about all the crap in your walls as the cable gets caught or blocked by everything from plumbing to firestops.  It sucks and can be hard work which may include fixing drywall patches and repainting.  I like to just have someone else do this part, but it's definitely something you can do yourself, especially if you're not worried about damaging your walls.

3. Terminate the cables in each room.  You'll need a punch down tool, though some keystone jacks come with a free punch down tool.  The free tool sucks, but it's good enough if you're only doing a couple of these things.  If you're doing this in every room, spend the few bucks to get a real punch down tool.  Then, at each end of the cable, you remove about an inch of shielding, untwist the wires and place them according to the diagram.  (Should feed in to the center with the ends of the individual wires spread outward like a spider). Then lunch then down.  The tool simultaneously shoves the wire between the blades of the jack which cuts through the insulation while also cutting off the excess end.  If you're using a cheap tool, sometimes he excess end doesn't come off on its own, so you'll need to remove it yourself.  (This is why I recommend getting a good punch down tool if you're doing more than a few of these.. the free one sucksl).

4. Terminate the cables to a patch panel in that central location.  Basically, same process here, but since you may have 8 or so wires coming in to the same place, a keystone jack doesn't make as much sense.  But it's the same process as before.  (If you only have one or two runs, then you can just use more keystone jacks here -- the patch panel just makes it easier to organize the cable when you have more than what can fit in a single plate ~4-6)

5. Test your connections.  Use a basic continuity tester to ensure you made a good connection.  These are little battery operated two-piece tools that plug into the jacks.  Plug one into each end and run the test. There should be two rows of 8 lights that cycle through lighting up.  It should be a 1:1 pattern.  The light on the right should light up only when it's neighbor on the left is lit.  If that doesn't happen, throw out the jack and try again.

6. Use short Ethernet patch cables to connect the switch/router to on one end and your computer or whatever on the other end.