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The library that you are using seems to rely on AVR specific interrupt and timer features for relatively tight I/O timing to read the echo signal and from there calculate the distance.
It will likely not work with the Galileo, since your code is executed as a program under Linux and I/O timing like what you need here isn't really possible.
If you would like to have the higher processing power you get with the Galileo (eg. for network, web service, xml, etc.) I would suggest that you add in a small Arduino or other micro controller that can do the measurements with your sensor, and then communicate with the Galileo board via UART, I2C or SPI.
I understand that it would be cool to have all code in one place and not have to deal with multiple units, but it will likely be very much up hill if at all possible to do it with just the Galileo.