In cosmology, there are three "things" in the universe: radiation, matter, and dark energy. At various points in the universe's history, each of these things have dominated the universe's evolution. We call these periods "radiation-dominated", "matter-dominated", and "dark energy dominated" (this last period also goes by other names since dark energy isn't well understood). The three are different because they scale differently with volume. In particular, if you have a particle in the box and double the size of the box, the volume goes up by a factor of 8, and the density goes down by the same factor. Mathematically we say $\rho(m) \propto a^{-3}$, where $\rho$ is the density, and $a$ is the so-called scale factor. With radiation, not only does the density go down, there is also a redshift. We write $\rho(r) \propto a^{-4}$. Finally with dark energy, if it's described by a cosmological constant, then it is a constant - its density does not change. $\rho(\Lambda) \propto a^{0}$.
The point of this is that during the periods when each of these things were the dominant component of the universe, the universe expands differently, and we can calculate how it is different. A different density of matter/radiation/dark energy for example would leave a different imprint on the CMB; conversely, measuring the CMB allows us to estimate the amount of matter/radiation/dark energy in the universe. From there, we can use relations such as this one (which in turn is derived from the Friedmann equations; as of time of writing it is the last equation in the section) to calculate the Hubble constant:
$\frac{H^2}{H_0^2} = \Omega_{0,R} a^{-4} + \Omega_{0,M} a^{-3} + \Omega_{0,k} a^{-2} + \Omega_{0,\Lambda}$
Here the $\Omega$'s are the present-day density of radiation, matter and dark energy, and $\Omega_{0,k}$ is the contribution due to the curvature of the universe.
The details are more complicated, of course. Here's another explanation of it aimed at a popular audience, or you could check a textbook on cosmology (however cosmology is usually taught at advanced undergraduate level, requiring an understanding of calculus and some knowledge of relativity).
