That is a Belden IBDN QCBIX1A4 BIX Distribution Strip with 4 pair markings. A datasheet can be found here. This product line was originally owned by Nortel, so you might see Nortel labels in your panel.
It's essentially a 4-pair splicing strip with 110 style punch down terminals - every pair of wires punched down on one side are extended to the other side of the strip.
The BIX Distribution Connector is a 25-pair connector. The
connector’s symmetrical construction allows termination
of cables on one side and cross-connect jumper wires or
BIX patch cords on the other. Each BIX connector is
equipped with 50 double-ended Insulation Displacement
Connection (IDC) clips for terminating plastic insulated
solid copper conductors without stripping. The connector
is built with two staggered rows of IDC clips enclosed in a
three-layer construction of fire-retardant plastic wafers.
Pair splitters on each side of the connector facilitate wire
Since it is certified for CAT-5e, you should be able to use it to extend ethernet to your rooms (and it should work up to gigabit speeds). Probably the easiest thing to do would be to get some solid (stranded wires don't work well with punch down blocks) CAT-5e patch cables, clip off the end (if you cut them in half, you can use both ends), then punch them down opposite the existing wires, matching the color codes. Then you just plug the other end of the patch cable right into your switch/router.
If you're lucky, everyone used the same color code, and it will work, otherwise, you might need to inspect your home wiring and the patch cable and match up the colors to make a "straight through" connection, i.e. pin 1 at the jack in your room, should go to pin 1 on the cable in your wiring closet, and so on. Here are the most common wiring color codes for structured wiring, but the installers (and/or patch cable manufacturer) may not have followed the standard:
You'll want to use a 110 punch down tool for this, a spring loaded tool works best:
(I'm not neccessarily recommending this particular tool - punch down tools range in price from $10 to hundreds of dollars for professional models. You're generally paying for longevity more than performance with the more expensive tools, for a few dozen punches, you won't see a whole lot of difference among the price range if you stick with known brand names)
But if you're only doing a few punches, a manual, non spring loaded tool would work:
There are other options, the BIX line also includes a CAT-5e modular jack strip that may fit into your existing panel, then you could punch down short CAT-5e cables to patch from the connector strip over to the modular jack panel. (or just move the blue wires from the connector strip over to the modular jack panel so you don't need to use short jumper cables)