If you draw a line in the plane with x horizontal axis and y vertical axis, then the rate of change is defined as how much the y portion of the line changes with an x change of the line, so a horizontal line never changes in the y axis no matter how much the x changes.
In mathematical terms, a line is defined with this equation:
y = mx +b
where m is the slope, or rate of change of the line.
A horizontal line has a slope of zero, so the equation reduces to:
y = b
that means that the value of y is fixed no matter how much x changes, therefore there is no rate of change.