Power is work done per unit time. Or the energy consumed or transferred per unit time-Joules per second.
The fan described in the post will consume 1100 Watts when operating at its rated speed when the supply voltage is 220V (that is such a huge fan!! 😁 ). This doesn't mean it always consume 5A. When the voltage is not 220V the device will not be consuming 5A or 1.1kW power, neither shall it be running at the rated speed at which it is designed to.
Take the case of an incandescent lamp. If the voltage is fluctuating, the glow of the bulb can be seen as fluctuating too. If the power consumption was always the same, there should not be any kind of fluctuations. That means, if the supply voltage was lower, it should have consumed more current than the rated current to keep the power consumption constant, which is impossible! The ratings on AC devices define that at the rated voltage that particular device will consume that rated current and so, 'that much' power.
Devices will work even without consuming rated power at a lower voltage, but might not always give an output which they are supposed/designed to give.
What exactly do they mean, when they say that the power output of a power plant is some kW ?
It means the power plant is able to meet a maximum power transfer rate equal to the described power output, beyond which it will fail. In other words, power output of a power plant is the maximum energy that the power plant can generate and transfer in a second.