A running track is a rubberized, running surface for track and field athletics. Competitive running races are run on it. A track meet is a meet where people race against each other on the track. How something is "going to run" means how something will proceed. It's clever that "going to run" was used for "going to proceed" because it has the word run in it, and a track meet is the subject discussed. The joke is playing on the word "run" and its different meanings.
Here is how the track meet is going to run . . .