There are two main reasons the story is structured as a letter. One is internal to the story, and one is external to the story. Let me mention first that a story structured as a letter is called an epistolary story or novel. An epistle is a letter. Stories written as letters are epistolary.
The first reason, the internal reason, is best stated by the anti-hero himself:
I have something important to tell you. See, the lady on the radio said, "... he wants to know the truth about Bangalore. [...] Mr. Jiabao wants to meet some Indian entrepreneurs and hear the story of their success from their own lips. [...] I offer to tell you, free of charge, the truth about Bangalore.
By telling you my life's story.
Balram is convinced his story is the epitome of the Indian experience as caste lines are blurred in the new India that focuses on big business and entrepreneurism. Internally, for the plot, the story is written as a letter (1) to cause explanations to a foreigner to be necessary and interesting and (2) to facilitate Balram's confessions and philosophical revelations that accompany them.
The second, the external, reason is that author Aravind Adiga chose the letter (epistolary) structure as a means of discussing and exposing social issues in India and, indeed, in the world, that have come to be in the more than six decades after India's independence (1947). One social issue in India is brought forward in the early pages of Balram's letter. He speaks to Jiabao of small Indian boys who run out into streets to sell American books on entrepreneurism to drivers of cars stopped at lights.
This highlights (1) that caste-free money making opportunities are highly sought by Indians and (2) that small boys are left to be uneducated because reforms to caste have not cut deep enough (these boys don't benefit from entrepreneurism as they are usually under the hire of an operator to whom they hand over the sales receipts from the passé American books).
One global issue brought forth, which appears painfully true and even more painfully disregarded, concerns the outsourcing of American work to India and the unhealthful lifestyles in America and other Western countries. This global issue is highlighted early in the story when Balram describes it for Jiabao:
[The] future of the world lies with the yellow man and the brown man now that our erstwhile white master, the white-skinned man, has wasted himself through buggery, cell phone usage, and drug abuse....