I need a simple text input area with the simple possibility of getting text input from user and add this variable to module for later use.

How do I set this variable to get set when user leaves textfield? Have tried several ways, but I'm not able to set variable on leaving field. When I need it later it says variable is not set.

In fact, text area (and other fields - Form field objects) object always updates associated variable when user leaves field. Associated variable name is defined in Form field object properties as "Run-time variable name", along with variable scope - current slide only or all module. CourseLab always stores current value of any Form field object in this variable and updates it anytime when the focus on field is lost.
Please make sure that you are not using non-latin characters and symbols in variable name - it may lead to variable malfunction.