A blue box gives you administrative access to the switching equipment by tricking the switching equipment into thinking it is receiving control plane information from another switch. It's like being able to send BGP messages to routers.
This means you can make long distance calls for free, but it also means you can route your call through whatever switches you want and lying about both the source and destination at each hop. Because wiretapping systems often rely on the switching infrastructure and with a blue box you control the infrastructure, you can make yourself invisible. It was God-mode for the phone network.
Fundamentally, a blue box is, along with the essential 2600Hz tone, simply a DTMF dialer, like what is built into touch tone telephones, only it used operator dual tone frequencies, not the same dual tone frequencies used by touch tone telephones. You can't do much at all with a blue box unless you understand how the switching equipment that detected those frequencies worked, and that was only documented in unpublished Bell Telephone Company and then AT&T manuals, then subsequently by phone phreaks who either learned by trial and error or somehow came across one of those manuals. Blue boxes stopped working due to the change to digital switching equipment.
A red box, which emulates the tone that tells a pay telephone or a long distance operator that a quarter has been inserted, should still function, if you can find a pay telephone that still accepts quarters.
This means you can make long distance calls for free, but it also means you can route your call through whatever switches you want and lying about both the source and destination at each hop. Because wiretapping systems often rely on the switching infrastructure and with a blue box you control the infrastructure, you can make yourself invisible. It was God-mode for the phone network.