Looking at the sky, the clouds, the moon and the stars really does make me feel clam and hopeful.
So, in this sentence how does the word "does" work? There's a problem without that word?
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"Does make" means exactly the same as "makes", it just adds some additional emphasis and carries the idea that you are pleased or even pleasantly surprised.
It is apparently known as the "emphatic do" and a common way to add emphasis and/or surprise, for example:
As another example, there was a popular British music hall song from the early 1900s entitled I Do Like to Be Beside the Seaside. The song is very 'theatrical' as was common with the style, and a great example of how "I do like" is far more emphatic than "I like".