To understand why the city was bombed in Fahrenheit 451 by Ray Bradbury, we have to put the novel into its historical context. It was published in 1953 in the middle of the Cold War, and its subject matter, although projected into a hypothetical future, is very much derived from the political and cultural situation of its times.
The book burning that occurs in the novel is an exaggerated reaction to the ever-increasing censorship that was taking place during the Cold War. The American public's fear of communist incursion allowed the rise of Joseph McCarthy and the House Un-American Activities Committee, which blatantly disregarded rights to free speech and other freedoms in its witch hunt for Communist subversives. At the same time, the even more terrifying prospect of nuclear warfare between the United States and the Soviet Union caused people to fear nuclear holocaust.
Bradbury wrote this underlying Cold War fear of nuclear devastation into his book. He does not specifically explain how the war started or why the opponents are fighting each other, but the threat of war is mentioned several times throughout the book as a sort of undercurrent of menace. For example, when Montag goes to visit the old man Faber, while they are talking together in Faber's home, Bradbury alludes to war preparations:
A bomber flight had been moving east all the time they talked, and only now did the two men stop and listen, feeling the great jet sound tremble inside themselves.
Just after Montag leaves Faber's home with the earpiece in place, he hears the government announce, "We have mobilized a million men. Quick victory is ours if the war comes." Faber then corrects the announcement, saying that if the government says one million men are mobilized, probably at least ten million have been.
When Montag is with the other bookish indigents in the wilderness, a jet flies by and Granger comments that "the cities won't do well in the next few days." Everyone is aware that the war is imminent, but no one can do anything to stop it. As mentioned above, this is the Cold War mentality that pervaded the culture back in the 1950s, and Bradbury wrote this into his book. The only explanation for why the city is bombed is that one side somehow provoked the other, but the specific cause is not mentioned in the book.