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The most direct evidence comes from supernovae observations in remote galaxies. Astronomers can use supernovas as 'standard candles' because the light we receive from them always follows the same pattern with time, allowing us to deduce how bright the supernova really is and by comparing it to how bright it appears on the sky, work out how far away it is.

Light from distant galaxies is also red shifted by the expansion of the universe and the red shift tell us how far way (back in time) the galaxy is and the rate of the expansion of the universe.

When astronomers calculate the expected amount of red shift for the supernovae based on how far away we know they are, they appear to be too strongly red shifted to account for the calculated expansion rate of the Universe. This means that the expansion of the universe is accelerating (which requires an energy to drive it). This energy is "dark energy".

There is also other, indirect, evidence for dark energy.

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Q: What is the evidence for dark energy?
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