Why is there a resistor in this circuit?
The 220 ohm pull down sets an operating point for the system. Without it, the fan would never turn off. With it, the fan will only turn on if the NTC resistance is low enough to pull the transistor base above 0.7V base-emitter threshold.
With a 220ohm pull-down, that transition will begin to happen when the NTC resistance gets below 2.6k ohms (about 55-57 deg. C). Some of the current will begin to flow in the transistor base, turning on the fan.
Adjusting the pull-down will shift the turn-on point:

increase above 220 ohm, fan will turn on sooner
decease below 220 ohm, fan will turn on later

It's slightly more complicated in the real world however.
First, the fan has some minimum turn-on voltage. A brushless 12V DC fan won’t light up with less than about 7V. So this will delay the turn-on point even more, to when enough drive is being put to the fan to satisfy this threshold.
Second, the transistor beta also influences this. A low-beta transistor will shift up the operating point compared to a high-beta one.