For the uninitiated, a "beige box" was just a piece of phone wire with a RJ11 jack on one end (the kind that plugs into the base of a phone) and alligator clips on the red and green wires at the other end. You used it by opening up the "demarc box" on the side of a building and clipping onto the corresponding terminals in the box, and using the line like normal.
What made this moderately useful was the emergence of COCOT (Customer Owned Coin Operated Telephones) phones. With a phone company payphone, the "magic" that made it a payphone was stored in the switch at the exchange, and so beige boxing one of those didn't do you any good. But the "magic" that made a COCOT a payphone was in the phone itself, so the line it was hanging off of was usually a plain jane phone line with full long-distance and everything available. So you'd find a COCOT at a strip mall or something, then drive around back to where the demarc boxes were, find the one for the payphone (usually by trial and error) and take it over. Now you could do all the war dialing or calling long distance you wanted. And hiding out behind a strip mall late at night, hidden behind a dumpster or whatever, you were fairly safe from being found by cops. In fact, of all the times we did this stuff back in the day, we never got caught by a cop in the act of using this setup.
Even better, some of the demarc boxes were rigged up internally to use an RJ11 connector, so you didn't even need the alligator clips. You could just open up the box, unplug the wire inside, and plug a regular phone in and have your way with it. :-)