am wondering if it's safe to send 12V to the Ground Pin on the arduino which I think is connected somehow to the usb port. Am afraid that it might burn something up. I found this sketch online -> 
Am a total newbie is this circuit safe?
am wondering if it's safe to send 12V to the Ground Pin on the arduino which I think is connected somehow to the usb port. Am afraid that it might burn something up. I found this sketch online -> 
Am a total newbie is this circuit safe?
here's the picture
Putting a short circuit from a 12V source to ground (0V) is liable to result in the release of the magic blue smoke.
However, limiting the current through a resistor, led and BJT (aka an electronic circuit) to a safe value will not.
I understand your confusion but like Jim Dearden says in the comments:
"you are not taking 12V to ground but the current from the 12V source through the LEDs (and their resistors) switched by the BJTs (c-e) to ground"
I think this particular section here is what's causing your confusion (correct me if I'm wrong)
Linking the grounds together like this is the correct thing to do, even if they are the ground points for different voltage sources.
There are multiple questions here that cover the topic of connecting grounds together I'll link a couple for you to do some extra bedtime reading:
Why connect two power supply grounds?
Connecting two grounds together
Connecting grounds together.... why? (Arduino forum)