What exactly does this mean and how is it used? Google tells me it means to run, and I’m assuming that’s not the same as courir. The latter I think refers to the physical act of running while the verb in question means to run as in a campaign, a show, something of that nature?
An affiche is a poster (of a show, a movie, a play), so the idiom
means a show is "holding its poster", i.e. is currently running somewhere and has enough success to keep going day after day.
There are also the expressions:
être à l'affiche
which just means a show is running now, without giving any indication about its success and
tenir/occuper/partager/quitter/... le haut de l'affiche ("to hold/occupy/share/leave/... the top of the poster")
which means that a show or a part of a show is the most successful of its kind, or a person has his/her name listed first on the poster of a show/movie, or is simply one of the leaders in some domain.