How did the presidency change during the progressive era?
The Presidency during the Gilded Age was largely made up of "caretaker" presidencies, where elected Congressmen and the President kept their hands out of economic interests and activities for the most part. This was called a "laissez-faire" economy, where there was little regulation and business was extremely powerful.
Once Teddy Roosevelt inherited the office from a slain William McKinley, he changed the nature of the Presidency to an agent of social and environmental progress, and as a regulator of that same big business to level the competitive playing field, and to give consumers some measure of protection from the most dangerous products and working conditions. With a few exceptions (the 1920s Presidents, for example) it's been that way ever since.
In general, the presidency as an office came to be much more important during the Progressive Era than it had been before.
In the late 1800s, the Presidency was very weak. Presidents did not exert themselves very much to try to affect policy. Congress pretty much ran the country.
Starting with Theodore Roosevelt, this changed. TR was the one who described the office as a "bully pulpit." "Bully" meant "really good." He believed the office gave him the chance to really push the people and Congress towards various ideas and policies.
Because of this, and because the Progressives wanted government to enact a lot of reforms, the Presidency got much stronger during this time.