How this is possible that the amount of radiation emitted to space from the polar region is greater than the amount of radiation that is received from the sun?
I don't know that this is true--do you have a source for this?
It is, however, certainly possible: the temperature in a region (along with its emissivity) is what determines how much thermal energy is radiated, but that depends on more than the amount of energy that region receives from the sun. Global weather circulation transports thermal energy (generally from the tropics toward the polar areas). In addition, water vapor is a greenhouse gas and the air over the poles tends to be drier than in the tropics, thus the absorption and re-radiaton of thermal energy back to the ground could be less near the poles. There are also other contributions to the earth's thermal budget such as the outflow of energy from the interior (much of it due to radioactive elements in the earth's interior which release some energy when they decay). This last is not a huge contributor but it is not negligible.
I agree with Paul. Drier air at the poles has less water. Water vapor captures about 90% of the radiant energy emitted by the Earth as a black body radiator. So less water allows more energy (primarily in the infrared region) to escape to space.
Separately, the decreasing ice pack, primarily in the Arctic during summer reflects less sunlight than in the past. But that is likely a smaller factor in net energy balance.
The basic principle is that incoming radiation is at wavelengths consistent with the Sun’s temperature while the emitted radiation is at infrared wavelengths consistent with the Earth’s temperature and water vapor plays a controlling role in the ratio. As water vapor like carbon dioxide has absorption peaks in the infrared.
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