"Does it mean there is a leakage happening in the attic from the edges of the skylight ?"

Check with the "wetness meter" owners manual and discuss with your builder. They may confirm your diagnosis.

"Flashing seems to be installed properly"

..............Wait until dry weather , test again.

and wait for the next heavy rain if it dries out before.

Or..... Get up on the roof with a hose. What "seems" may not be reality.

IMO get someone who knows about roofing and skylights to check it out . I garee new drywall finish could be reading damp but we have no idea. I agree the easy way to findo ut if the skylight is leaking is to run a hose there.. I do not agree with millworkman as to not worrying but you need a pro

The drywall sanding is done few weeks back and it’s painted few weeks back already. Also if touch the meter at other places it doesn’t beep. I am trying to call to some local roofing companies to see if they can come and do an inspection.

Were the drywall wet you would be able to tell, it would be fairly obvious.

"Also if touch the meter at other places it doesn’t beep."

Are you checking other seams and places where joint compound was applied, or are you comparing the area you are concerned about to random sheetrock surfaces?

Thanks kudzu9 for the pointer. I was checking on the corners of the wall around skylight. Did some research and found the touch less moisture meters create a magnetic field and when they hit a conducting material (like water or metal), they start beeping showing moisture. In my case it was the cornerbead metal strip hidden behind the drywall compound.
When I moved the meter up away from corner, it stopped beeping.
Thanks everyone for your help.