How was the Presidency of Teddy Roosevelt the epitome of American expansionism, Progressivism, and the modern presidency?
Teddy Roosevelt was our first Progressive President. Before him, the movement lacked focus, and could not break into national politics in a meaningful way to effect change. It seems strange, then, that he was also an imperialist, that is, imperialism and progressivism seem to be opposing philosophies.
Almost all of our imperialist acquisitions at the beginning of the 20th century happened in Roosevelt's term. This includes the Panama Canal zone and establishing American hegemony in the Caribbean. He sent the Great White Fleet of battleships around the world on a mission of "peace", flexing American muscle and projecting American power around the world.
He also managed to get Congress to go along with progressive reforms by using his popularity (The "bully pulpit") to pressure the legislature to back his ideas. The modern presidency evolved from TR's actions at that time. With the exception of perhaps Ronald Reagan, I don't know if anyone has come close to mastering politics the way Roosevelt did.