How and why did the federal minimum wage in this country start?

When he took office in 1933, one of President Franklin D. Roosevelt's goals was to establish a federal minimum wage.
Regulations establishing a nationwide minimum wage were part of his National Industrial Recovery Act of 1933. However, the Supreme Court ruled on May 27, 1935, on what was called "Black Monday," that a minimum wage was unconstitutional, overturning the regulation.
Roosevelt pushed for a minimum wage because he wanted everyone who worked to earn enough money to live on. He fought for this as part of his New Deal for the American worker. After the disaster of the 1929 stock market crash and the onset of the Great Depression, Roosevelt felt that the average American needed greater protection to ensure a decent standard of living.
Roosevelt believed that paying working people enough to live on was the morally decent thing to do; he also hoped that higher wages would help lift the United States out of the Great Depression. In addition, Frances Perkins—his pick for Secretary of Labor—would not agree to the job unless Roosevelt was firmly behind a minimum wage law.
While some argued that a minimum wage would hurt business, Roosevelt argued that a business that could not pay its workers a fair wage deserved to fail. As he put it in 1933:

No business which depends for existence on paying less than living wages to its workers has any right to continue in this country.

In the 1936 presidential campaign, which he won in a landslide, Roosevelt campaigned on a promise to enact a minimum wage. In 1937, the Supreme Court reversed its 1935 decision and supported a minimum wage law in Washington State. This paved the way for Congress to pass the first minimum wage legislation, called the Fair Labor Standards Act, in 1938.
This legislation established a minimum wage of $0.25 per hour; in addition, it enacted child labor laws and a 40-hour workweek, with time and a half paid for overtime work. Ever since that time, the United States has had a federal minimum wage. Roosevelt called the Fair Labor Standards Act one of his most important legislative victories, and it has remained popular ever since.