That's one of those structured media boxes, probably from one of the box home improvement stores. Personally I hate those since they aren't very deep and are a bitch to work on when full. What I believe you have are the telephone bridge modules for that structured media box. You'd have to figure out which lines are which. Is green supposed to be telephone only or is blue supposed to be telephone only. My guess is that the different colored cables were meant to show one is telephone and one is data, but someone punched them both down to telephone bridge modules.
You'd need to pick up a 110 punchdown block (or the Data bridge modules for that structured media box), remove one or both of the telephone bridge modules (assuming you don't need either of them), then punch down the wires to the new 110 block. After that, you can use patch cables to plug into a ethernet switch or router. Alternatively, less optimal setup would be to remove the telephone bridge modules and terminate the wires to 8P8C CAT5e (or CAT6 if it is CAT6 wiring) connectors. Then you can plug the wires directly into an ethernet switch or ethernet router.
Also you'll need to check what each wire terminates to in each room. I assume there are CAT5e keystones and possibly CAT3 RJ-11 keystones at each faceplate. Could just be all CAT5e keystones. You want CAT5e keystones so you can plug your computers/game consoles/streaming media players/TVs/etc directly into the jack.