I work for an online editing and proofreading service where we see many, many letters of motivation. Please see my profile for more information. I won't revise your whole letter here, but I will show you the changes I would make to your first and second paragraphs (which I have combined into one).
To The Selection Committee:
I am writing to apply to the __ Exchange Programme. I am from the Netherlands. I was born in Nijmegen where I went to primary and secondary school. I moved to __ almost two years back to attend __ __ __. RSM was the perfect choice for me. I have completed all my first-year courses and all courses of the first trimester of the second year with a grade-point average of__. I now want to be part of your programme, to gain new experience, get to know myself and other people better and expose myself to a new environment.
Your introduction is basically sound, but I have condensed it and tweaked the sentences to make them an easier read. I have also corrected some common, though not serious, errors like the salutation. Never begin with "Dear Sir/Madam" unless you want your recipient to throw out the letter before he or she has finished reading it. In the United States most applications are decided by a selection committee. If you have reason to believe an individual will be the one making the decision for this program, find out the name and preferred title of that person (from the programme website or by making a phone call) and use it. Also business correspondence salutations always end in a colon.
Avoid stating "My name is" in your introduction since your name should be on your CV and the rest of your application as well as the signature and the return address. I omitted the "Let me get started...", the bit about living without parents and the fun and hard work of the first year because they are all unnecessary filler. Your letter is a lot stronger without them. You also should, in the first mention, spell out the name of your university and, if it's not an embarrassment, include your grade-point average (or the European equivalent). "Sufficient grade" is not a familiar phrase to me. For a U.S. program (note the spelling difference) your sentence should be, "I have passed all my first-year courses as well as all the courses of the first semester of my second-year." In the U.S., the only common usage of trimester is in describing pregnancy! Good luck and take care.