From Wikipedia.org:

The T-shirt evolved from undergarments used in the 19th century, by cutting the one-piece “union suit” underwear into separate top and bottom garments, with the top long enough to tuck under the waistband of the bottoms. T-shirts, with and without buttons, were adopted first by miners during the late 1800’s as a utilitarian covering for their hot environments.

T-shirts, as a slip-on garment without buttons, originally became popular in the US when issued by the Navy during or following the Spanish American War. These were a crew-necked, short-sleeved, white cotton undershirt to be worn under a uniform. It became common for sailors and Marines in work parties in hot, tropical climates to remove their uniform shirts, thus utilizing only the undershirt. It is possible that the Navy uniform boards first discovered the usefulness of the T-shirt by observing the practice of dock crews.

Named the T-shirt due to the shape of the garment’s outline, they soon became popular as a bottom layer of clothing for workers in various industries, including agriculture. The T-shirt was easily fitted, easily cleaned and inexpensive. For this reason, it became the shirt of choice for young men and were mass produced in various colors.

By the time of the Great Depression, the T-shirt was often the daily garment worn when doing rough farm or ranch chores, as well as other times when a working necessity called for a covering made from lightweight fabrics.

Following World War II, it became common to see veterans wearing their uniform trousers with their T-shirts as casual clothing. The practise became even more popular after Marlon Brando wore one in the movie, A Streetcar Named Desire. The actor’s fame finally achieved status for the T-shirt as a fashionable, stand alone, outer-wear garment.

Thus began the popular T-shirt trend we see today.