How does the redshift - distance graph show the rate of expansion of the universe at every moment in time?

From the best fit chart curve you describe

vertical: z (redshift) horizontal: d (distance)

charts of two other functions can de derived, which I will explain later in this answer. The two functions are:

The function $a(t)$ is called the "scale factor".

https://en.wikipedia.org/wiki/Scale_factor_(cosmology)

The function $H(t)$ is called the "Hubble parameter".

https://en.wikipedia.org/wiki/Hubble's_law#Time-dependence_of_Hubble_parameter

I do not know how much math you have, so I will try to keep the math simple. Trying to answer your question with no math would be very difficult.

The time t it takes for light to travel from the distant object to the observer is

$t = d/c$,

where c is the speed of light.

z has a relationship to the expansion of the universe.

$a = 1/(z+1)$

$a(t) = frac {1} {z(d/c)+1}$

The data you have for the best fit of the graph $z(d)$ could be used to make a graph of $a(t)$.

I do not know if you understand derivatives, but for now I will assume you do. (If you don't, please let me know, and I will try to explain it in detail.)

The Hubble constant, $H_0$, is the value of $H(t)$ at time

$t = t_{now}$.

$H(t)$ is the expansion rate at time $t$. The way in which $H(t)$ changes with time is related to how $a(t)$ changes with time.

$H(t) = frac {da(t)/dt} {a}$

On the $a(t)$ chart, $da(t)/dt$ is related to the tangent of the curve at time $t$. If you take two points along the tangent, say ($a_1$,$t_1$) and ($a_2$,$t_2$), then

$ frac {da(t)} {dt} = frac {a_2 - a_1} {t_2 - t_1} $

The above is intended to answer your two questions. I would now like to explain that the concept of "best fit" by itself is very ambiguous, and astronomers use other mathematical techniques to obtain the most accurate useful models of the universe, including the functions $a(t)$ and $H(t)$. There is an equation commonly called the Friedmann equation which is derived from general relativity.

https://en.wikipedia.org/wiki/Friedmann_equations#Detailed_derivation

This equation relating a(t) and H(t) has five parameters (constants) whose values correspond to a model of the growth of the universe from its beginning to the present (omitting two very early phenomena: "inflation", and the dynamics of primordial nuclear fusion, and also omits the mass of neutrinos).

https://en.wikipedia.org/wiki/Inflation_(cosmology)

https://en.wikipedia.org/wiki/Big_Bang_nucleosynthesis

The astronomical data is used to "best fit" the choices of the values of the five parameters by one of several choices of criteria as to what determines the intended meaning of best bit.