Why did a relatively simple burglary like the break-in at the Watergate turn into a major political scandal and bring down a president?

Because the break-in was carried out by political operatives working on behalf of President Nixon, and Nixon himself was involved in trying to cover up his administration's connection to the burglary efforts.

Context/details:

The first break-in by burglars was in May, 1972, as persons connected to the Nixon reelection campaign broke into the Democratic National Committee office in the Watergate complex of buildings in Washington, DC. They were planting wire-taps on the Democrats' phone lines, and also stole copies of documents. When the wire-taps didn't work properly, they broke in again (in June) to try to fix the surveillance devices, but they were caught.

The formation of the "plumbers" actually preceded Watergate. They were a "special investigations" unit that the Nixon White House had set up in 1971 to stop the leaking of classified information. Members of this group were those then later involved in the Watergate break-in (among other activities).

"Deep Throat" was the name given to an inside source at the FBI that gave information concerning Watergate to reporters Bob Woodward and Carl Bernstein of the Washington Post. Years later it was revealed that Deep Throat was Mark Felt, who at the time was Associate Director of the FBI.

Nixon's role in Watergate was especially in his efforts and those of members of his staff to cover up what had happened.

Ultimately, the Watergate affair brought down the Nixon presidency. He resigned in order to avoid impeachment. And the whole affair made Americans more distrusting of government.