You are not invisible.Your IP address is recorded within the email header.If you are using your ISP connection for illegal purposes and the cops get interested,your IP address will lead them to your front door after they identify the ISP and obtain your private details from such.Other than law enforcement,the closest anyone will get is the name of your ISP and your general location.
Caveat:I'm not an IT expert,those here who are will no doubt fill in any missing pieces.