You would remove the green cable from that phone distribution module, and connect the wires to a 'data distribution module' - basically a punch-down patch panel. This affordable Leviton item is available from Home Depot:
You would then use a short 'patch cable' to run from that patch panel to the router or separate switch.
Ideally, that patch panel would be mounted to the rear of your enclosure. That linked one is made by Leviton, and designed to snap into a Leviton bracket. The bracket snaps into the holes in the back of a Leviton enclosure. You don't have a Leviton enclosure, so you'll need Velcro, foam tape, or metal screws; the holes probably won't line up.
Go to Home Depot or Lowes, to see what's offered locally. I think Home Depot has a selection of Leviton items, and Lowes, IIRC, carries OnQ hardware.
There are many other ways to approach this. A keystone patch panel, mounted in the enclosure, would be another way to do it. I think adding male RJ45 connectors to the green cables would be much more expensive, and tricky, for the novice.
You'd also need to buy a punch down tool to add the wires to that punch-down patch panel, probably $15 from HD/Lowes, cheaper online. Keystone RJ45 connectors include a free small plastic punch down tool that works fine for a handful of connectors.