It's called a “bib,” and it doubles your shirt's chest fabric, ensuring that anything visible under your tuxedo jacket is bright white, not see-through. Only wear bib-front shirts with a tuxedo and bow tie.
What is the best shirt to wear with a tuxedo?
Generally, we suggest solid white twills and broadcloths for tuxedo shirts. Twills are more opaque and have more shine than broadcloths, lending them a more formal appeal. Twills also drape better under tuxedo jackets. Broadcloths feel smoother and more modern, but are generally more sheer than twills.