Respected commenter Mitch explains what is going on with rising Asian SAT scores:

It's a mix of things. 

First, the 2005 changes made the math subject material easier, but an actual high score became harder because there were fewer hard problems and the impact of "unforced errors" became higher. Kids who were bright but careless could get a high score because the occasional unforced error was wiped out and more with their performance on the high difficulty questions. That last sentence describes whites more than Asians.  

So since 2005, the ability to nail every question and not make unforced errors—something that drill does, indeed, help—has been rewarded, whereas the number of creatively difficult problems is 0 or 1 per test. This hurts white students, on average, more than Asian students just by personality trait, and then the Asian tendency to drill for this test gives even more of an advantage.  

The reading test has been made unequivocally easier. I'm not sure what you mean about some reading questions being moved to the writing section. This is not true. The writing test is a near-exact replica of the old English Composition Achievement test, or the English Writing Subject test. There were no changes to it at all from a content perspective—they just changed the type of essay prompt and reduced the number of questions. 

Certainly, the easier reading test makes it easier for Asians to get high scores. The writing test rewards attention to detail above all 

So the test changes play a part, both in how they reward the traits more likely to be in asians over whites, and then in the Asian prep ritual—which really has to be seen to be believed. I teach in these schools, and the kids are in prep taking tests for 2 years. Even the ones who aren't getting super high scores are getting better scores, and that's bumping the average up. 

Then there's the fact that Koreans, Chinese, and Indians are immigrating here in huge numbers, which is presumably offsetting the once larger percentage of Filipinos and Tongans.  

I'm assuming you were only looking at US students, right? [I don't know — it's not obvious from the College Board documents.] Koreans in Korea are taking the test as well, and there they study 40-60 hours a week, instead of school. The prep schools there buy copies of the most recent tests from students and use the tests to prep their students (something that's frowned upon here, although not technically illegal). The kids learn how to write essays by rote, and have whole essays memorized (use this essay for "change", this one for "education", and so on).

There's a general lesson here. Our society is constantly changing and bending procedures in hopes of Closing the Gap between the more feckless and the more feckful racial groups. Yet, perhaps unsurprisingly, the more feckful tend to better exploit these changes for their own benefit. 

For example, the changes in federal regulator attitudes toward zero-down and low-doc mortgages that George W. Bush announced in 2002 in the name of fighting racist redlining at his White House Conference on Increasing Minority Homeownership  poured hundreds of millions of dollars into Angelo Mozilo's net worth, most of which the SEC allowed him to keep even after fining him. But the net worth of the median Hispanic and black is now lower after mortgage follies.

The changes in the SAT in the last decade were mostly due to a single man, University of California chancellor Richard C. Atkinson. The UC system is the College Board's biggest client, and the UC was told by California voters in 1996 that racial preferences were now a violation of the state constitution.

Atkinson held an inflated notion of his expertise in the field of psychometrics. "When students asked me about IQ testing, I frequently referred them to Stephen Jay Gould’s book The Mismeasure of Man, published in 1981; it is a remarkable piece of scholarship that documented the widespread misuse of IQ tests," he wrote in an essay explaining part of his motivations, although he disingenuously left out all mention of the 800 pound gorilla in the room for UC, Proposition 209. The state legislature's Latino Caucus had threatened to cut the university system's budget unless they could figure out a way to cheat on Prop. 209 and get more Latinos admitted.

Now, Atkinson wasn't a complete cynic. He was more of a fool. He thought he could kill a whole bunch of birds with one stone. For example, he wanted the SAT to be more of an achievement test than an aptitude test because of his quasi-Gouldian views on IQ testing:

My views are similar to those of Alfred Binet, the French psychologist who, in the early years of the last century, devised the first IQ tests.  Binet was very clear that these tests could be useful in a clinical setting, but rejected the idea that they provided a meaningful measure of mental ability that could be used to rank order individuals.  Unfortunately, his perspective was soon forgotten as the IQ testing industry burst onto the American scene. 

This is like saying:

My views are similar to those of Nicolaus Copernicus, the Polish astronomer who, in the middle of the 16th Century, devised the first heliocentric system. Copernicus was very clear that heliocentrism reduced the number of crystalline spheres and epicycles necessary, but rejected the idea that we could explain the motions of the planets without the Music of the Spheres. Unfortunately, his perspective was soon forgotten as Kepler, Newton, and Einstein burst onto the astronomical scene.

Atkinson's goal was to make the SAT less of an aptitude test and more like the SAT II Subject Tests, which are more achievement tests. (UC long demanded applicants take both the SAT and three SAT Subject Tests, a high degree of overkill.) Since we've all know about how blacks and Hispanics are obviously oppressed by the white man's IQ pseudo-test, the thinking went, this would ameliorate UC's little problem with the Latino Caucus by Closing the Gap.

Except, it didn't work out that way because blacks and Latinos tend to not only have less aptitude, but they tend to be lazier about achieving academically. Also, it made the new three-part SAT 3.75 hours long, which may tend to mentally exhaust blacks and Latinos. The winners from these "reforms" turned out to be Asians, who, on average, work hardest. 

Of course, the UC schools already had plenty of Asians. So, a few years ago, UC, having turned the SAT into something like the SAT Subject Tests, announced it wanted to stop requiring the SAT Subject Tests as now being redundant. This sensible reform, however, outraged the Asian-Pacific Islander caucus in the California legislature. The more tests parents have to remember to sign their kids up for and pay for tutoring in how to beat the tests, the better Asians do versus everybody else.

One thing to keep in mind is that elite private colleges don't really seem to want more Asians (at least if they're not going to pay list price. Foreign Asians, who don't qualify for financial assistance, are increasingly fashionable with colleges.). The more Asians score high on the SAT and get inflated GPAs by taking a lot of AP courses, the more the super-prestigious private colleges, which can and do use quotas for admitting blacks and Latinos, appear to discriminate against Asians.

Why they discriminate against Asians has been speculated about, but not, so far as I know, studied in any truly illuminating fashion.

Do they not donate as much money to their alma maters? My vague impression is that Indians, with their ancient tradition of alms-giving, are pretty generous when they have a chance to get their name put on an academic building. (I recently took enthusiastic part in a three minute standing ovation for an Indian gentleman who was the chief donor for the new library at my son's high school. It's a really nice library and it was all paid for and build by the week before my son started school there, so it didn't cost me a penny, so clapping my hands sore was the least I could do.) Chinese benefactors? Maybe not so much ... I don't know. This is the kind of thing that colleges have no doubt studied in intense detail, but their findings are Top Secret.

Or maybe too many Asians is considered uncool by high school students. For example, UC Irvine has long been heavily Asian, but it never seems to climb in coolness with kids. 

Or maybe the elite colleges just don't believe the high SAT scores being recently recorded by Asians. Who knows? Obviously, elite private colleges know what their motivations are for requiring higher paper credential from Asians, but they aren't telling, and nobody seems to be asking.