I assume you're talking about a microphone like this one: Electret-mic. If so, you will need first a proper interface to connect it with Galileo. The interface should include an amplifier because the microphone's output is an analog signal in the order of milivolts.
The microphone's output can be connected to one of the analog inputs the Galileo has, and you can read the analog input from your code. You can store the read values in a file for example, and then you can open the file, and read the digital values that represent the analog signal.
In order to reproduce it, you will need a DAC (Digital-Analog Converter) to get the analog signal again. This analog signal can be connected to a speaker and you should hear the sound. Again, a proper interface for the speaker might be needed. Galileo doesn't have a DAC so you will need to use an external one. An interesting method to make a DAC is the following: Resistor ladder. It only uses resistors but the quality might not be the best.
There are a lot of projects like this one for Arduino, I recommend you to look for them. You might get ideas for the software and hardware part of your project.