Why did the Republican Party split during Taft's administration?
It was transitioning from the progressive party to the rich northern businessmen's party
From its founding in 1854 until Theodore Roosevelt left office on 1908, the Republicans were socially progressive. They began as the anti-slavery party and their first elected president, Abraham Lincoln, did in fact end slavery in the US. Theodore Roosevelt was a "trust buster" who took a hard line against the opportunistic millionaires who had thrived in the Gilded Age of the 1890s.
But William Howard Taft, Roosevelt's hand-picked successor, was a lot more business-friendly, as were his successors (after an interruption by Wooodrow Wilson, a Democrat) Warren G. Harding, Calvin Coolidge and Herbert Hoover.
By the time Roosevelt's cousin Franklin (a Democrat) took office, the two parties had effectively swapped roles. the Democrats were the party of labor, and the Republicans were the party of management and Wall Street.