My guess would be that the old man is telling this story to his daughter as if it's just about a bunch of guys.  But when he says, "My boss fired him," it becomes clear that the father who's telling the story was in fact the driver of the car who decided to play hookey from work.  It sounds like the others in the car were workmates whom he usually drove to work, but in this case took with him to the beach for a day off.

light = traffic light

boardwalk = a typical elevated board sidewalk at very commercialized beaches, where people stroll and gain access to the amusements and restaurants etc.  There's a very famous one in Atlantic City, New Jersey.  There's a famous 50's or 60's song Underneath the Boardwalk, which is the singer's favorite spot for making love.

get a tan = the reason many people go to the beach   -   to catch some rays.

I can't comment on how funny it is.  When he was young, he and his workmates decided to take the day off.  As the instigator, he was the one who got fired, but it was not a complete loss, since he got a tan.  I suppose the thing that makes it funny is that the old man pretends to be telling his daughter a joke, but it fact it's a true story about himself.

 - A.