It's finding out the difference: My room is brightly coloured.

I have 3 sentences:
1.The sung is shinning bright.
2.They greeted us bright.
3.My room is brightly coloured.
Why is the word “bright” used in sentence 1 and 2? Why not ‘brightly’ as in sentence3?

Because it is wrong. These are correct:

1.The sun is shining bright(ly).
2.They greeted us brightly.
3.My room is brightly coloured.

why is it wrong? how can I use them?

You must use an adverb (“brightly”) to modify a verb or an adjective. “Bright” is an adjective.

You will hear some people use “bright” in conversations but that doesn’t make it right.