Let's take it a step at a time, Arje.
I am writing to you to apply
--that's pretty obvious. Better stated would be, "I am applying..."
for the Master’s Degree
--oddly, it's Masters Degree (not possessive) or, in short, a Master's (possessive).
in English and American Studies in which I am particularly interested in.
--double "in" in there. Drop the last one. And a comma after Studies would clarify that you are interested in the entire subject, not just some specific elements in which you are interested.
My aim is to gain more knowledge, become more efficient, share experience
--the last of the list should have an "and": "... efficient and share experience..."
from native professor’s .
--professors. It's a plural, not a singular possessive. And no intervening space before the period/full stop.