If you use "jeho", it means František is looking for someone else's car (Matěj's, for example).
We use "svůj" (svá/své...) when the possessor is the subject. Here, František is looking for František's car. Hence "své".
- I'm looking for my car - Hledám své auto, because I + my.
- František is looking for my car - František hledá mé auto.
- I'm looking for his car. - Hledám jeho auto.
I'm pretty sure all this information is in the Tips&Notes which you should read prior to starting lessons.