Temperature over the last 400,000 years has been measured in Antarctica by extracting a core (a cylinder a few inches in diameter) of ice about two miles deep. The ice is made up of year after year of snowfall, so it contains a sample of snow from most of those years.

Snow is water is H₂0, hydrogen and oxygen. But both H and O come in two varieties, heavy and light. Heavy hydrogen (1 proton and 1 neutron) weighs twice as much as normal hydrogen and it's call deuterium. (Heavy Oxygen has two extra neutrons, but is only 18/16 as heavy as regular Oxygen). In both cases the ratio of heavy to normal atoms in snow varies with temperature. So by looking at (mainly) the ratio of deuterium to normal hydrogen in the ice they can determine the temperature when it snowed.

Since temperature changes most at the poles, the change was actually about 10°C, and Hansen divided that by two to give a rough but decent estimate of the average temperature change of the earth as a whole.

For a possible source of the 10°C value see ORNL which mentions that the overall amplitude of the glacial-interglacial temperature change is ~8°C for atmospheric temperatures above the inversion level and ~12°C for surface temperatures.