If you want a simple Internet-connected temperature logging application, for example, you can just use the built-in temperature sensor - this is all the hardware you need, self-contained and ready to go. You don't even need the breadboard - just solder the two battery wires to the board.

But you've also got the option of using a breadboard if you want to prototype different hardware connected to the ESP8266.

I've left the shield off this one, for the photo, so you can see the chip inside.

(When I went to take this photo I couldn't find a 9V battery snap anywhere, so some improvisation was needed.)