A Statesman by definition is a person with an understanding of the principles of government who is actively engaged in conducting the 'business' of a government as well as introducing and shaping public policies. Teddy Roosevelt fits this description because he rolled up his sleeves and had a 'hands on approach' when it came to governing and administering policy.
1. As the Police Comissioner of N.Y.C he was aware of the indiscretions many police officers were engaged in (drunkardness, gambling, sloppy in appearance, etc.) so he walked the streets meeting up with these men and with his demanding presence told them to basically get their act together. His tactic was simple...speak plainly and to the point. For example, he would say something like 'have you taken a good look at yourself lately? 'don't you realize that you are in a position to command the respect of the people in the neighborhood?
2. Roosevelt viewed his presidency as a 'stewardship of the people' meaning that he believed that the president should control and use power in the best interests of the people. This belief was at the core of his policies which led to major reform legislation in the U.S. The Pure Food and Drug Act and The Meat Inspection Act of 1906 was directly related to Roosevelt's shock as he learned about the horrors of the Chicago meat packing industry by way of the muckraking novel The Jungle. After reading the novel Roosevelt went and saw for himself the horrid conditions of the industry.
3. Roosevelt also believed he had a responsibility to the future generation of Americans. His was concerned that the natural beauty of the nation was being stripped away by industrialization therefore he used his political powers to protect millions of acres of land from development. The National Forest Service was designed to protect and maintain public lands for the public.
Roosevelt was and is considered the first president to believe that the presidency must take an active role in government affairs. While advisors and assistants have their role in the Executive Branch, Roosevelt really enjoyed working as the president same as a man laboring on the farm.
Roosevelt was the first real diplomatic leader and voice of the American Empire. McKinley helped to establish the empire through the Spanish-American War, but he didn't live long enough to lead it, being assassinated in 1901.
Roosevelt on the other hand has a whole list of imperial accomplishments, among them the Great White Fleet, the Roosevelt Corollary, the acquisition of the Panama Canal Zone and the beginning of construction, winning the Nobel Peace Prize for his role in settling the Russo-Japanese War, and waging war in the Philippines against Emiliano Aguinaldo's rebellion there.
Since his time, it has become typical of US Presidents to take a leading role on the world diplomatic stage, but Roosevelt himself was a pioneer.
You could say he is a statesman because he spent a fair amount of energy on foreign affairs. In addition, some of this effort was spent on things that, you could argue, helped the world.
The most obvious example of this came in the episode of the Russo-Japanese War. In this case, Roosevelt got the diplomats of the two countries together and negotiated a peace between them. For this, he won the Nobel Peace Prize.
You might also argue that he was a statesman for the ways he worked to make the United States stronger. You could cite the Roosevelt Corollary to the Monroe Doctrine here.