What type of connector is this, and how to connect cable into it?
It's an insulation displacement type connector. It works because when the black wire is pushed into the "jaws" of the metal knife edges, it cuts through the insulation and makes an interference contact with the conductor inside the wire.



Normally a tool is used but you may be able to get a blunt instrument and push the black wire back into place. I'd probably solder it afterwards to ensure the connection is good BUT make sure the plastic case can withstand the heat without distorting.