Answer
When we think of the instantaneous rate of change at a given point we want to know how much the function changes per unit length of the narrow interval containing $a$ in the limit when the length of that interval tends to zero. This is exactly given by the expression for the tangent slope.
Work Step by Step
The slope of the tangent at point $(a,f(a))$ is given by
$$m_{tan}=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$$.
When we think of the instantaneous rate of change at a given point we want to know how much the function changes per unit length of the narrow interval containing $a$ in the limit when the length of that interval tends to zero. This is exactly given by the expression for the tangent slope.
