"Du haut de [qqchose]" (literally 'from the top of [smthing]') usually indicates watching/managing something from a higher position after achieving the something referred to.
A certain disdain usually leaks from these words: the subject is standing above and looks down on the following part of the sentence.
Du haut de mes 25 ans d'expériences, je suis à même de ... Intending to say: I am mature enough, so I am obviously capable of doing ...
In your example, the words are written by someone else, thus there is no contempt, and the author just praised the longevity of the show.
In this case, you could re-write it: "Après plus de 15 saisons, [...]"