Why does a horizontal line have a rate of change of zero?

Because since the line does not increase in y values and stays constant throughout where it is placed on the graph, and it does not change, its rate of change is zero.

Since slope is essentially rise over run, a horizontal line (which doesn't change y values) will have a slope of 0/1 (the one can be a different number depending on the specific graph, this is just an example). It rises by 0 and runs by 1. 0/1 is equal to 0. You should know this.