We measure the temperature of far-away objects by investigating the radiation (mostly light) they emit, and we can do that right here from the Earth. Scientists long ago figured out what temperature goes with what colour and what brightness, so if they know the brightness or colour, then they can easily determine the temperature. If you do not know how far away the object you are looking at is, then you can only use the colour to figure out what its temperature is. This is the case for many stars. In this way, the surface temperature of the Sun has been found to be about 5500 °C (i.e. 9900 °F or 5770 K).
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