Why do we feel wind, even though it is “invisible”?

Answer

Wind is nothing but moving air. When those air molecules flow against our skin, we feel it.
You can definitely see that wind, or rather that sky. If the sky were completely transparent, we would also see a black sky with stars during the day. But the sunlight is spread by the molecules, and because this mainly happens with the blue wavelengths, the whole sky turns blue, to the extent that it drowns out the stars.
So we see those air molecules for exactly the same reason we see any other object: because they reflect light. Only, of course, you cannot see individual molecules, because they are too small.
You can sometimes also see wind itself, if there is enough dust in the air, or if the air is very humid, such as with a cloud. That’s why you see a tornado spinning effectively.