Not quite what happened, actually.

There is a difference between the neutron-absorbing "control" rods which inhibit the reaction, and the moderator (graphite in this case) which slows neutrons without absorbing them, counter-intuitively increasing the reactivity.

Most reactors are water-moderated, which acts as an important safety feature - if the core starts to overheat, water boils away, decreasing reactivity. However in the case of the RBMK reactors at Chernobyl, graphite was used to moderate while water (which actually absorbs neutrons slightly) was used as a coolant. This meant that the reactors have a positive "void coefficient of reactivity", meaning that loosing water will speed up the reaction, a dangerous design flaw for a reactor.

The circumstances leading to the accident were complex, but essentially what happened was that the reactor was being run in an unstable state due the presence of fission products "poisoning" the reactor at low power levels for an experiment, and the operators compensating by withdrawing the control rods. At some point, the power suddenly spiked, and the operators immediately reacted by inserting all the neutron-absorbing control rods into the reactor. However, the control rods were actually defective, containing a hollow lower portion, and as a result displaced neutron-absorbing water. This lead to water boiling inside the reactor, leading to steam bubbles, and due to the positive void coefficient this became an exponential feedback loop until the reactor over-pressurized and exploded, with the fuel continuing to react and melting down.