There are a few different bits of evidence. Hope I get them reasonably correct,
Firstly light from a distant object that is moving away from you looks more red. Because the wave length is being stretched. The light from distant galaxies is observed to have shifted red from the expansion of space in between us. The farther they are the redder the light because there has been more time to expand.
Edit: if the universe were contracting then the light would be shifted blue.
Secondly it was predicted that if the universe used to be hotter and denser , then there would be some left over energy that would have cooled and ‘stretched’ if the universe had expanded. And they have found that background energy fitting the prediction.
Thirdly different types of elements like hydrogen and helium need different conditions to be formed. And the balance of types of elements we have detected also fits a hotter denser period followed quickly by an expansion and cooling.
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