One of the most basic methods of analysis is the ancient "method of exhaustion": in its most basic form, if $x$ is a number with the property that it's smaller than every positive number and bigger than every negative number, then $x$ must be zero.
Take a moment to digest exactly what this statement is saying, and to convince yourself of the obviousness of that fact.
This obviously generalizes to other numbers than zero. And it can be applied to more general ideas, especially when they can be quantified.
An example of a more general idea that can be quantified is that of a linear approximation to a function near $a$. You say you already know this so maybe you want to skip this section, but I want to explain it in terms of exhaustion.
We usually insist on it being exactly right at $a$, so it must be of the form:
$$ f(x) \approx f(a) + m (x-a) $$
$$ x^2 \approx 1 + m (x-1) $$
The value $m$ quantifies the approximation in some sense: we usually call it the "slope".
We note that $m=1$ is a better approximation than $m=0$, because:
- For $m=0$, the error is $x^2 - 1 = (x-1)(x+1)$
- For $m=1$, the error is $x^2 - (1 + (x-1)) = (x-1)x$
note that for every value "near" $1$; e.g. every $x \in (0, 2)$, the error for $m=0$ is worse than the error for $m=1$.
It turns out that every $m < 2$ is too small to be the slope of the best approximation, and every $m > 2$ is too big. Thus, the best approximation is at $m=2$: $m=2$ is exactly the right number to be the slope of the "best linear approximation".
And indeed, if we check the error, we get $x^2 - (1 + 2(x-1)) = (x-1)^2$, which is a smaller error for values of $x$ sufficiently close to $1$ than $m=0$ or $m=1$, or any other value of $m$.
Now, what might the phrase "rate of change of a function at a point" possibly mean? If we wanted the change between two different points, or the "average" rate of change over an interval, that's easy to understand.
But what does it mean to be the rate of change at a single point?
It is hard to give a "direct" definition that works in every case we might consider, but we can combine the method of exhaustion with our intuition of rates of change to give a definition of this phrase: any value that is less than the slope of the best linear approximation is clearly too small to deserve to be called the rate of change. Similarly, any value that is greater than the slope of the best linear approximation is clearly too big. Thus, we define the rate of change of a function to be the number in the middle: i.e. the slope of the best linear approximation, a.k.a. the derivative at that point.