The author explains immediately:
He had perfect scores — on his SAT, on three SAT subject tests and on nine Advanced Placement exams — and was ranked first in his high school class of 592. An admissions officer who reviewed his application to Harvard called him “the proverbial picket fence,” the embodiment of the American dream, saying, “Someone we’ll fight over w/ Princeton, I’d guess.”
Wikipedia says
Picket fences are a type of fence often used decoratively for domestic boundaries, distinguished by their evenly spaced vertical boards, the pickets, attached to horizontal rails. Picket fences are particularly popular in the United States, with the white picket fence coming to symbolize the ideal middle-class suburban life.
The author is roughly saying that this student was an ideal candidate--the kind of student every other student wishes they could be.