John Cheever wrote many short stories in what was referred to as the 'New Yorker style'. Why might a writer have a style? What is your writing style and why?

The former question's fairly simple to answer. The latter, maybe not so much, considering ones perception of their creation is almost always skewed.

The most fundamental and important reason for adopting a style is one that lies at the heart of our society: something those in business would call a Unique Selling Point (USP), something that satisfies an itch in its audience so that they keep coming back.
In the case of Mr Cheever, I get the feeling that his USP is a certain accessibility. The Lexis is simple, but still carries a depth of meaning: e.g. "for Lawrence it would be an easy step from the coast of Portugal to the tyranny in Spain" (taken from Goodbye My Brother). Here the character's pessimism is conveyed through a simple metaphor, and all one needs to understand it is to have some knowledge of world news at the time of this story's initial publication. It's through aspects like this that his stories become enjoyable even for the common man. One doesn't need a Shakespearian upbringing, or a life spent in reading. All they need is eyes, a civilised life, and half a brain.

Plus there's nothing like calling a character a "son of a bitch" to break down pretences!

It is a similar wish for honesty and accessibility that I try (keyword being 'try') to fulfil with my own writing. I have no sovereign right to a reader's attention, so I should not attempt to make them work too hard to enjoy what I write.