Help connecting cat5e cables for home networking
The yellow cables are probably Cat5e, which has 4 pairs of wires. (The beige cables are probably Cat 3, which might only have 2 pairs of wire). A single phone line only requires 1 pair, so as you can see the other pairs are just twisted off.

You definitely can reuse the cable for ethernet access, but you will lose the phone jack obviously.

Here are some things you should know about working with ethernet cabling:


Unlike phones with are relatively low-bandwidth, ethernet cables need to be in good condition with the pairs twisted around each other properly and with properly terminated connections. The mess you show is fine for phones but will cause lots of interference with ethernet data. If you want to reuse the cabling for data you need to cut back the cable all the way to the yellow sheathing and start over with new CAT 5e terminations. The proper way to do it is to terminate all the cables at a patch panel, but you could also just crimp a regular RJ-45 on the end and connect them that way. You cannot just twist the wires together and expect to get a good signal. Also, at the other end of the cable (in the room with the jacks), you will need to do the same thing: cut back the cable to where it's clean, and start over with new RJ-45 jacks. 
Also unlike phones (and household electricity), you cannot split a cable into 2 cables just by joining the same color wires. If you have more than two devices you want to connect, you need to use an ethernet switch, which is a small appliance that will properly route the data packets between 2 or more devices.