You do not need additional resistors/capacitors on the bread board. All the components shown in the linked schematics are already included on the printed circuit board you've got from Adafruit.
VDD/GND is connected to the supply voltage. Connect only voltages within the given limits and of correct polarity!
Starting at the top of the left side, there are two power pins, VDD and GND that are used for powering the chip. These should be 2.7V to 5.5VDC. There's no polarity protection so make sure you get the wires in the right polarity! Higher voltages will give you more power so if you want that full 2.8W you need to give it 5VDC.
R+/R-, L+/L- are the audio input. Those are for differential signals. Connecting to the phatDAC (does not output a differential signal) requires you to simply connect R- and L- to GND and R+ / L+ to the respective R / L output of the the phatDAC.
There are four pins for the audio input signal. For right channel, R+ and R- and for left channel, L+ and L-. These are differential inputs. If you are connecting this to a device with differential outputs. just connect the + and - pins as indicated on that device's outputs. If there is only one differential reference, connect L- and R- together and tie that to your differential reference. If you are using every-day single-ended audio signal, connect L- and R- to ground, and L+ and R+ to your signal.
SDL/SDR are the shutdown pins, they have pull-up resistors on the board. As long as you do nothing (not pulling them down to GND) the device is enabled.
You can turn off the amplifier for each channel separately. To turn off the right channel, connect SDR to ground. To turn off the left channel, connect SDL to ground. By default these pins have pullups to VDD so both channels are on by default!
G0/G1 are the gain settings. For now they do not need to be connected, just use the DIP switch to select the proper setting.
The amplifier has 4 voltage gain settings. You can set the gain either by switching the pins on the onboard DIP switch or connecting the G0 and G1 pins to ground. By default the two gain pins have pullups to VDD so turning a switch 'on' is the same as connecting the Gx pin to ground.
When you set the gain, you'll want to probably set the gain so you have the highest peak-to-peak voltage that is still lower than VDD, if your signal gets higher than VDD you'll get annoying clipping sounds. The gain you set depends, then, on what voltages you have to work with, as VDD and also the signal level. You may want to start with 6dB and increase the gain until you get the signal levels you like.