How can I hack my intercom system to buzz people in remotely?
3 possible solutions:


Connect the 2 contact for the button to a relay and control that relay from the arduino (example) But the kit you have doesn't include a relay.
Use a transistor to trigger the switch -- I would not recommend this option as your first project.  It would be easy to damage the intercom and/or your arduino.
Use the servo in the kit to physically press the button.  You will need to figure out the mechanical linkages but you don't need to attach wires to the intercom.


For a beginner project I would probably go with the third option.