The author as far as I can read your quotes is correct. This is not that easy but try to understand. In western type reactors like PWR and BWR (Google them) also in Soviet/Russian VVER type reactors (they are very similar to PWR) the core is filled with water and as the heat from the fuel assemblies heats the water it becomes pressurized, that is why they are called "Pressurized water reactors", now water is also a neutron moderator which means it slows down the fission neutrons as they fly through the water atoms. Remember that for U235 to fission it needs the so-called thermal neutrons which are slower in their speed (kinetic energy) than fast neutrons, if there was no water almost all the neutrons from the fuel rods would be fast because there would be nothing to slow them down. U235 fissions much less with higher kinetic energy/fast neutrons. The reasons for this you have to either ask in a different question or study through google.
Now back to Chernobyl, for reasons beyond the scope of this answer the RBMK-1000 is a different type of reactor, a so-called graphite moderated light water cooled reactor. In this reactor graphite moderates neutrons, it slows them down and reflects them. This means that in an RBMK reactor the U235 fission reaction can happen even if there is no water present because there is graphite in the core, now normally there is also water in the core pipes running through them cooling the fuel and producing valuable steam for the turbine. Water as a substance absorbs neutrons more than graphite, so water is a better neutrons absorber than graphite while both water and graphite can moderate neutrons, aka slow them down. Now as the water flow through the RBMK core was stopped, more and more water turned to steam. Steam is much less dense than water so the neutrons absorbing properties of water were eliminated so now there was only graphite left, and the control rods which had to absorb excess neutrons were almost all taken out, this means that the reactor can now boost itself up in power and since neutron increase happens very fast the operators couldn't simply stop the increase as it was running very fast, the power was doubling each split second.
The control rods in the original RBMK took about 18 seconds to fully insert from a fully withdrawn position, so much too slow to stop a runaway chain reaction.
You might know that this is what happens in a nuclear bomb, sort of, a critical mass is reached and there is no neutron control, they are allowed to run wild, and that is what is actually wanted in a bomb but not in a power reactor
The miniseries is actually kind of correct on this issue because the accident itself was a sum of operator error and design specifics of the RBMK reactor.
There are certain specific details that make the RBMK very unstable at low power but I'm not sure how much information are you capable of understanding so for now let me stop here.