What is this black box for hooked to security system?

Short answer, 555U, It's a transponder interface module for a remote start.
Long answer, Many newer cars made within the last 15 years have some type of anti-theft immobilizer built in. One way to prevent a thief from "hot wiring" your car is integrate the key with a transponder chip that is programmed to the immobilizer ECU. When you insert your key into the key hole, there's a transponder detector coil around the key hole. Transponder detector coil detects the transponder and send that unique code to the immobilizer ECU. When the right code is received, the immobilizer ECU will allow the engine to continue to run.
When you make a key with the right cut but with the wrong or un-programmed transponder code, the engine will start and run for a few seconds then the engine dies. Many cars will allow you to try a few times before it locks you out for upwards to an hour or so.
When installing a remote start, the installer is basically electronically hot wiring your car. In order for the immobilizer ECU to let the engine continue to run after started, it needs to have that transponder code. This is where that black box come to play.
Underneath the green tape around that black rectangle plastic on the PCB has a coil. The red plug should have red (power) black(ground) and blue (remote start activation). The white plug has 2 black wires (Transponder sender)
When remote start is activated (ground on blue wire), The relay energizes and activate the transponder detection coil under the green tape. The transponder code is then sent through the white plug to another coil that's placed around the key hole. The factory transponder detector coil detects the transponder code from the DEI's transponder coil. The immobilizer ECU therefore detects that correct key is being used to start the car.