Why are their traffic cones at the barn?

Because most riders "steal" some at some point to use as a training aid. Traffic cones are used as a cheap alternative to poles for weaving exercises, as halt markers in arenas that don't have dressage letters, to teach novice riders where the letter X (the center of the arena) is, etc.

So every barn will have a few around - sometimes actually acquired, uh, legitimately. (Come on, we've all done it).