It is based on the basic concept of taking a single electrical component and making it easy to use. Usually the electrical component is an Integrated Circuit. Integrated circuits have pins on them which can do a multitude of things, but you usually have pins to supply power, providing a ground, receiving an input and an output.
A Breakout Board breaks out these pins onto a printed circuit board that has its own pins spaced perfectly for a solder less breadboard, giving you easy access to use the integrated circuit. There are different types of breakout boards but most of them are for different types of sensors, for example accelerometers, ultrasonic distance sensors, RFID tag sensors, temperature sensors, pressure sensors, and they even have seismic breakout boards for sensing dinosaurs footsteps.