The answer to why to use a sprite-sheet has already been answered in this questionquestion.
How you use a sprite-sheet in your game is quite simple. Usually you have a class that keeps track of the elapsed time and displays another frame once a given threshold has been reached (eg. if 1/25th of a second has passed or similar).
How to display a new frame depends on your underlying graphics API. You might have to copy the new pixel data to a pixel-buffer, or modify UV coordinates of a quad, so that it shows another portion of the sprite-sheet.