It is a story of a boy and his younger brother going across the US, on their way to a massive NES Competition in Los Angeles. Along the way they practice and play popular NES Games.
At the end of the movie the 3 finalists compete at Super Mario Bros. 3.
At that time the game hadn't been released yet. Putting SMB3 in the movie was a way to get people into theatres, as well as promote the movie. The movie wasn't a big success, but probably did much better than it world have, by having SMB3 in it.
Here are some screenshots from the movie of the SMB3 scene.
Click on each image to view it in its full size.