I've produced sound by using the MAX98306 amplifier, but any solution will work. The actual sound comes out of a USB sound card, so any solution that takes a 3.5mm headphone jack will work.
Take a look at the assembly instructions for the MAX98306 breakout. +5V and GND will come from the spare wires attached to the Cape in that assembly step. I strongly advise using either jumper wires with female pins installed, or crimping them yourself, as being able to disconnect this makes assembly MUCH simpler. L- and R- will connect to the sleeve on the headphone plug. L+ will connect to the tip, and R+ will connect to the ring. Each headphone plug assembly may be different, so just check the datasheet for the selected plug.
I tagged the left channel wires with a green heat-shrink. For the wires, I used pre-cut and crimped jumper wires, which was expensive, but saved me the effort of crimping the female jumper connectors.
Do not forget to place the included jumper block over one of the sets of amplification select pins. I wasted a good deal of time troubleshooting the audio setup for having missed that.
On each speaker, you will see + and - terminals. Either use a narrow gauge speaker wire, or simply solder some red and black 18 gauge wire to the speakers. The kit has screw terminals, as shown in the above picture.
Plug the headphone plug into the output jack of the USB audio card, and plug that into the BBB. Test it again, and there should be a 'bong' as the game starts up. The speakers do seem to emit several seconds of static until the sound card is initialized by the kernel. I have chosen to call this 'normal'. There is a 'shutdown' pin which could be wired to a spare GPIO to inhibit this during startup, but this is an exercise for the reader.