In this case, the verb screen means: to examine (people or things) in order to decide if they are suitable for a particular purpose
Example: In the US, when you go to the airport, you have to go through security screening by the TSA.
In this example, "Tick off" means to count off. More often, however we hear it used in a different way which means to make someone mad or irritated. Example: If my father does not let me take the car to town, it will tick me off. or when my father did not let me take the car to town, I really got ticked off.