In simple terms a breakout is something that takes a grouped set and breaks it into its constituent parts. For example, you may have a number of signals in an audio snake. A breakout is the point where the snake separates into its individual lines. Similarly you could have multiple channels of audio on a single optical cable, but have a device that acts as a breakout box to separate them into individual inputs and/or outputs. A multi-out option for keyboards and drum machines is a type of breakout box. There are many other examples.