The universe is expanding in all directions. We can detect this by the redshift of electromagnetic radiation from other galaxies.
We detect a larger redshift in galaxies which are further away, and I understand it is from this that we infer that the expansion of the universe is accelerating.
Why do we infer this, and not that expansion was faster in the past (when the light from those galaxies was generated)
I read this question but I didn't understand the physics equations used in the answer.
The size of the universe is given by a scale factor, normally written as $a(t)$, that is a function of time and is calculated by solving Einstein's equation for an isotropic homogeneous universe. Once we've calculated $a(t)$ we can differentiate it wrt time to get $\dot{a}(t)$ and use this to calculate the recession velocities.
The scale factor is a function of time, but as we look farther away we are also looking back in time so we can measure $\dot{a}(t)$ over a range of times and not just at our present time. Then we can compare experiment with the results of our calculation.
Now, when we calculate $a(t)$ and $\dot{a}(t)$ there are some adjustable parameters that we need to feed in by hand. One is the average density of matter, and another is the cosmological constant, $\Lambda$, i.e. dark energy. If we set the cosmological constant to zero and do the calculation then our calculated values for $\dot{a}(t)$ do not match what we see by looking at type 1a supernovae. In order to make the calculation fit the experimental data we need to use a non-zero value for $\Lambda$. This in turn means that $\dot{a}(t)$ is increasing at late times, which means the expansion is accelerating.
The point of all this is to answer your question:
Why do we infer this, and not that expansion was faster in the past (when the light from those galaxies was generated)
The point is that we aren't randomly making up the rate of expansion. We get it from General Relativity. The way $a(t)$ and $\dot{a}(t)$ change is dictated by GR and our supplied values for the density and $\Lambda$. The expansion can't have been faster in the past than we calculate unless there is something fundamentally wrong with either GR or our assumptions about the early universe.
I read this question but I didn't understand the physics equations used in the answer.
Let me offer a simplified explanation. The origional Big Bang cosmology asserts that the universe is expanding, which if true means that the proper distance $D$ between any 2 points is increasing. Since you can think of this expansion as "space itself expanding," then the rate of at which 2 points separated by $D$ are changing is proportional to $D$ itself and thus the "velocity" $v$ (or rate of change of the proper distance $D$) is $$v\propto D\rightarrow v= H D.$$ This is Hubble's law, with the constant $H$ relating $v$ and $D$ known as Hubble's constant due to it's empirical discovery/verification by Edwin Hubble.
We detect a higher redshift in galaxies which are further away, and we infer that the expansion of the universe is accelerating.
The redshift is related to the velocity $v$ between the source of the light and and receiver/observer and can be computed by the (relativistic) doppler shift given knowledge of the value of $v$. If we lived in a universe that had a constant expansion rate than the velocity (as observed via the redshift) would simply be proportional to the distance $D$ by Hubble's law.
The problem is Hubble's constant $H$, isn't really a constant but depends on time i.e. $H\rightarrow H(t)$. In addition, because it takes time for light to travel, the further away an object is the further back in time we are effectively looking. Therefore looking at the redshift of objects at very large distance $D$ from us will give us information about what $H$ was in the past. What we find when we look at far objects is that $H$ was smaller in the past, i.e. that the expansion of the universe is accelerating with time.
