There may be many reasons why Stanford wanted to give him a chance.
Perhaps they saw something more in him? Perhaps he had a part time job his last year, so that he couldn't spend so much time perfecting his score? He can now focus 100% on Stanford without worrying about money (most of the time), and can probably more than keep up with the "elite".
The article also mention that students who come from families who can afford more, spend money on better high schools, SAT preparation, private teachers etc. to get a high score. In that case, a 1300 might not be so bad, when others need so much extra help.
Here's a current chart showing how many students in the most recently reported high school class (students who graduated from high school in 2008) scored at different levels as a sum of scores from the critical reading section (200 to 800 standard score points possible, scores rounded to the nearest ten) and from the math section (same scoring) when they took the SAT for their best individual total score for one sitting.
Although you are not an American, you have a good understanding of what else college admission committees look for besides high test scores alone. Your other comments mention several issues that admission committees claim to look at when deciding on applications.
Stanford's most recently reported interquartile ranges for SAT section scores of enrolled first-year students are here:
Please note that a few years ago, when the student mentioned in the newspaper article was admitted, the ranges would have been a bit lower.