I haven't seen this movie myself, so I may be missing some context, but this is what I get out of it:
"Matinee days" are days when the theatre has afternoon performances. Afternoon performances aren't done every day because they get smaller attendance during the day - most people prefer to go at night.
In my experience from movie theatres (rather than stage theatre) "matinee days" are the ones where afternoon tickets are sold at a discount (movie theatres tend to run the movie throughout the day regardless, but afternoon showings are not as popular, so some days they have a "sale" on those tickets to encourage attendance at those times.) For example, where I am it's usually Tuesday where the afternoon shows are cheaper.
Sidney is suggesting that Steve Dallas is dodging his responsibilities on matinee days. Playing the fool to get out of doing his job so he can go watch a movie (or stage play; again, I haven't seen the move so I'm unsure what type of theatre he'd likely be going to, but the idea remains the same regardless.)