Understanding a circuit diagram
I tried connecting the outer strand to the ground of the electret mic
  and the inner strand to the other terminal. The blue wires are
  supposed to go the speaker. I connected one to the ground of the
  beaglebone and the other to an analog pin.


Two problems (more than likely)


Your electret mic needs a polarizing current to activate the circuit inside it.
Don't connect the speaker wires to ground - you may (have) damage(d) the amplifier


Try a 10 kohm resistor to positive supply for biasing the electret microphone and try using an audio transformer for pulling off the signal from the amp (assuming it isn't now damaged).

It could easily work without the audio amp just by biasing the electret has mentioned above. I have no-idea what the input is on the beaglebone so I can't comment on where you need to AC couple it via a capacitor and/or dc bias it to midrail.