The application process used to be tedious manual filling up of hardcopy forms and mailing them over. Nowadays, however, the application process can be done entirely online and is strongly encouraged over the hardcopy way of applying.
Applications typically open during March and close in April. Dates may differ from year to year and from country to country, so do please check the MOE website regularly. You will be asked to mail in a few certified true copy documents, and those are also listed out on the website.
I have heard that on average over a thousand applicants apply for the scholarship every year across all levels. After which, a few hundred will be selected for a selection test in June. Of these numbers, around a hundred applicants will be called up for a selection interview in August. Finally, around 5-30 will be offered the scholarship. I have no idea how true these statistics are. I may be completely off tangent, so it's up to you to use these facts discerningly.
There are no special tips or tricks in applying that I can offer apart from the usual "study hard, do well" advice. They will be requiring your school grades for the past two years. For example: if you are applying for the Form 3 intake, they will be asking for your Form 2 results as well as your Form 3 latest results (e.g. term 1 or mid-year results). So do perform consistently well in your school academics. It is not necessary to be the top student in order to be called up for the selection test. Nor is it compulsory for you to be in the top class or anything like that. I have known many scholars who weren't straight A students. I myself didn't score straight As for my UPSR. Bottom line is: as long as you perform consistently well, you'll probably be fine.
Another aspect that they will probably consider is your extra-curricular activities. As with all scholarship boards, they want to know that they will be splurging their money on worthwhile causes. Who wants an academic robot who only knows how to score strings of As on their report card but will look like a little lost lamb after you snatch their textbooks away like you snatch candy away from a baby? I'll paint you a picture: let's hypothetically assume that Malaysia as 1000 primary schools. Each of these schools will have a top student. And all these top students apply for the scholarship. What is going to set student A apart from student B would be his/her extracurricular activities. So, while you can, go and explore new things. Go learn how to play the flute, go join your school's badminton team, go push your limits with math olympiad, go set up an anime club and cosplay as Naruto. Whatever it is, go and try things out. And when you find a love for something, it will show on your extracurricular records because you will enter more competitions, you will practice harder, perhaps win a few trophies etc. If you're one of those last minute applicants and are fretting over a non-existent extracurricular record: what has been done is done. You can't change your life as it was anymore. I encourage you to just go ahead and apply. There is no harm, really, in trying. I too, have friends that have gotten the scholarship with almost non-existent records.
There are three components to the test: English, Math and an aptitude test.
1) English Test
If you come from an English speaking family or background, this test should not pose too much of a problem for you. Basically there will be a comprehension (pemahaman) paper as well as a composition (karangan) paper. The test standard will vary accordingly depending on which intake you are applying for. For mine way back in 2005, I was asked to write a letter to my sister recounting a recent birthday party I had. It helped that I did have my first birthday party just a few months before the test, and that I do have sisters...but that's another matter altogether. In other words, I got lucky with my test.
For non-English speaking students: during the months leading up to the test, I suggest you start practicing the language. Start speaking it to the people around you. Perhaps even start a diary to improve your writing skills. Personally, I came from a Chinese type primary school where the hardest English UPSR paper questions were like: The watermelon is ___. (A) Big (B) Round (C) Triangular (D) Square. Until now I still find it tough to pick between A and B. Technically though, it should be spherical, and not round, but I digress. My point is: that it does not matter what background you come from. I believe that MOE understands that Malaysians do not get enough English exposure (now with them reverting back to Malay as the teaching language.)
Most importantly, regardless of how excellent you think your language is, the best way for anyone to prepare for this test is to read. Read read read. (something other than the Twilight series, please)
2) Math Test
Many Malaysian applicants find the math test the most difficult component of the selection test. I wouldn't know because I honestly cannot remember my math test. From what I hear however, I think I can confidently say that the Malaysian schooling math syllabus would not be enough for you to sit for the test comfortably without pulling out all your hair and leaving a bald patch at the end of the test. I have friends who bought Singapore math textbooks a year in advance to prepare themselves for the test. You can do the same, I guess, though personally I think it's a bit of an overkill. Just go online and look at the math syllabus for Singapore schools. Learn up new topics that your school has not covered. And if you're really kiasu, you can start learning topics in advance. For example, you are applying to enter Sec 3, go learn up the Sec 3 syllabus.
Also, bring a calculator if they asked you to! You will definitely be needing it if they said so.
3) Aptitude Test
Also widely known as the IQ test. Contrary to popular belief, they will not be testing on general knowledge. So no need to go cramming your vocabulary or the newspaper into your heads.
This test is purely visual. Basically they will have a big picture with various shapes or patterns. At the bottom right hand corner, there will be a missing piece. There will be five answer options below (A,B,C,D,E) for you to choose from. And all you have to do is to choose the correct missing piece that will correspond to the picture! Sounds easy? Only if you have all the time in the world. The key aspect to this test is how well you manage your time. There will be around 50-60 questions for you to do only in 20 or so minutes. Also don't be fooled by the dead-easy questions in front and start being complacent. The questions get harder as you progress.
If you have made it this far, congratulations! This will be the last hurdle you have to cross in your journey of applying for the scholarship.
You will face a panel of three interviewers in a cold cold hotel room. So do bring along a jacket. In your letter inviting you for the interview, MOE will probably list out a few items you are required to bring along - such as documents or school transcripts etc. Don't forget them, though I doubt forgetting them will immediately strike you off their list. Wear smart casual attire. No flip flops, tank tops, baggy bermudas, etc.
There are no fixed questions for the interview. Perhaps they will ask you questions pertaining to your selection test results, or your extracurricular activities, or even why you hate dogs. For me, they asked me to describe my primary school. I'm sure they asked more, but I cannot remember. The difficulty level of the questions will probably be adjusted according to your age group. I highly doubt they'll ask a 12 year old about the Eurozone debt crisis nor will they ask a 17 year old to describe their favourite movie. Or maybe they will?
Just be yourself - that the best advice I can give.