How do I activate my Cat5e+ connections int my apartment?
Each of the coax wires, green wires, and blue wires represent home runs from this box to your room plates.  Respectively carrying TV, telephone, and ethernet service.  (I agree with Ecnerwal's observations)
Presumably that black coax hooked to a splitter is your network input.

You can use this in two sorts of ways;
1) put your cable modem and a router/switch here, and hook the cable modem ethernet output to the switch, and wire the switch ports to the blue patch jacks. 
If you want wi-fi, you can put it here or on a drop that is optimally located.
2) you can use one of the coax drops to run to the room where you want to put your modem and switch(s).  Back feed the ethernet from that room to a switch in this box, all other rooms would be downstream of that switch.

WRT to TV.  You can add an additional multi-line coax splitter (maybe that horizontal coax thing is one) to feed the other coax drops.  There may be two in a room to allow a DVD or satellite receiver source signal to be distributed to other rooms.