Hi pcribbin,
There is no SoftwareSerial on the Galileo, but contrary to the Arduino Uno, you can use the hardware serial port on pin 0 and 1 (called Serial1 in Arduino IDE) and still have debug informations to the computer via Serial as on the Arduino Uno.
All you need to do is remove the software serial code on line 13 and line 19 and change mySerial to Serial1 and of course move the connections to pin 0 and 1. I can't remember which is RX and which is TX, but you can look that up.
That should make it work!
/Thomas