Why does a car cabin fan need a big resistor?
The resistor is used for voltage reduction to the DC motor in your car fan. The power dissipated in the resistor can be calculated from \$ P = \frac {V^2}{R} \$ or \$ P = I^2 R \$. The resistor needs to be able to radiate this heat away without getting so hot that it melts nearby plastic components or setting itself on fire - hence the size.

The fans in your house are AC and so a capacitor can be used to reduce the voltage without any power being dissipated. A capacitor has an impedance (the resistance to AC) given by \$ Z = \frac {1}{\omega C} = \frac {1}{2 \Pi f C}\$. The current through the capacitor is 90&deg; out of phase with the voltage so the power dissipated is zero.