what does watt specify in induction stove
Watt specifies the power of the device.
Power is defined as energy per unit time. So watt is how quickly your device consumes energy.

If it is an efficient device it will convert more of consumed electricity into heat. So given two equally efficient devices the higher watt one will produce more heat faster and can therefore cook faster than the less watt one.

And the third part of your question has already been answered above. Yes, higher watt device will consume more electric power.