So I just discovered this strange-looking box in my apartment closet. What’s it for?
My apartment complex claims it’s “pre-wired for high speed internet,” which I had assumed meant it had ethernet ports in the rooms, but it only appears to have the standard two telephone jacks (that are too small for ethernet cables) in each room. Maybe this thing or the abundance of telephone jacks has something to do with that claim?
that is where the phone company wiring connects to the phone wiring of your building. you can test if you have phone service by plugging a phone directly in the jack that connects to your wiring. it makes it easier to see if your phone troubles are with the phone company or your phone wiring or equipment. unless you pay the phone company for the service you are responsible for all the phone wiring after that point.
your guess may be correct on the prewired issue meaning lots of phone jacks for connecting DSL. may also refer to having cable TV jacks. you will have to ask the building manager what they mean.
As JohnPost said, it looks like it’s a demarc (demarcation point). The “end of the line” for the telco’s responsibility. It also looks like an excellent opportunity to steal free phone connectivity from the neighbors. 
This happened to me. In the early nineties I was renting in a basement apartment in Mississauga. The guy renting the upper part of the house seemed to have a different car in the driveway each week. I never spoke to him much; he was often away, and I had a job to go to and a life to live.
One day I came home to a phone bill of over $400, laden with calls to Germany. I know no-one in Germany. I called the phone-company ‘police’, who said they’d send someone over to check the lines. If there was nothing wrong with the lines, they would assume I’d made the calls, and expect me to pay them–the usual story.
But lo! At the demarcation point, there was an extra set of wires connected to my line! They led upstairs. The phone company investigeted further, and I eventually learned some of the details.
Seems the guy upstairs was using my phone line to do business with. Not only was he making calls on it, he was giving the number out to people in South America and receiving calls when I was at work.
Shortly thereafter, he vanished and stiffed my landlord for 5 months’ rent. I heard later that the police picked him up in Germany… for speeding. On the Autobahn. I didn’t have to pay for any of the phone calls to Germany, though.
This demarc does not give access to other units’ wiring. The OP’s apartment has seven phone jacks and no apparent alarm system. The place is pre-wired for as many as four telephone lines at each jack. (Were they expecting people to run call centers from home?)
A typical alarm would be plugged into the RJ31X jack at top left - this lets the alarm panel take control of a line to dial out regardless of whether or not the line is in use.
To get access to other units’ phone lines, you’d need to find a similar but much larger panel called an MPOE (Main Point Of Entry) somewhere in the basement.
