There's nothing contradictory there. It means that a body cannot emit more or less than it absorbs. An object with low absorption will heat up less in the same time as an object with high absorption, and correspondingly the less absorptive material will radiate less energy than the more absorptive material.
Precisely! And that's why the shiny steel ball doesn't glow bright orange like the rusty ball.
OK. So you're saying that in a vacuum (where heat loss is limited to radiant losses) a rusty steel ball will cool much faster than a shiny steel ball. Where can we see experimental confirmation?
The main exception to this is shiny metal surfaces, which have low emissivities both in the visible wavelengths and in the far infrared. Such surfaces can be used to reduce heat transfer in both directions; an example of this is the multi-layer insulation used to insulate spacecraft.
https://en.wikipedia.org/wiki/Thermal_radiation#Surface_effects
EDIT: That's why the space blankets are shiny, to reduce heat loss.
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