First of all: To use boinc, I'd recommend installing it through the software repositories, not from a script. (Simply search for boinc in the Ubuntu Software Centre. You can get boinc manager there, too).
There are many advantages in using the software sources instead of downloaded scripts. It's more secure and you'll get automatic updates through the update manager, for instance.
I never used this script and don't know if it's meant to open any windows or just print to the console its run in. To run it and see what's printed, open a terminal (Ctrl+Alt+T or search Terminal in the Dash). You'll need the path the script is located in. Let's say its
/home/fossaw/Downloads. You'd type
/home/fossaw/Downloads/boinc_7.4.42_x86_64-pc-linux-gnu.sh in the terminal and hit enter, which runs the script.
If something is printed to the terminal that sound like there is a permission problem, run
sudo /home/fossaw/Downloads/boinc_7.4.42_x86_64-pc-linux-gnu.sh. It will ask you for your password and hence run the script with root privileges.