What did Senator Pittman believe was the reason for Republican opposition to joining the League?
Key Pittman was a senator from the state of Nevada that served as Chairman of the Committee on Foreign Relations and president pro tempore. Pittman was a staunch supporter of the League of Nations and Woodrow Wilson. The question you ask is about a Senate speech that Pittman gave in November of 1919. It would be very difficult to answer this question without understanding some background on the political players of the day.
The president, Woodrow Wilson, was a second term democratic president and was not in very good health, having suffered a stroke. The League of Nations was the hallmark of his Fourteen Points and, in fact, the only part of his legacy that was included in the Treaty of Versailles. The Republicans controlled both houses of Congress and were led by Wilson's chief political rival Henry Cabot Lodge. Lodge did not want the United States in the League of Nations because he thought it would compromise America's ability to make decisions unilaterally. Wilson was not at all interested in compromising on any parts of the Treaty and insisted the Congress ratify it unconditionally.
Pittman insinuates in his address to Congress that the Republicans are using the issue for political gain. The implication is that they are being deceitful to the American people and playing on their fear of being involved in further global conflicts. In doing so, they help their own political interest to take the White House in the 1920 election and maintain power in both houses of Congress.