In order to gain access to the secret rainbow levels, you have to jump on the flagpole when the timer shows the correct time. If done correctly, you'll hear the original NES world-cleared-jingle followed by a firework and a rainbow. When you return to the world map, the entrance to the secret rainbow level will appear at the entry point of each world on the far left. Secret rainbow levels contain no enemies but hundreds of coins and eventually golden flower power-ups.