When shooting video what is actually happening when the shutter is "off"?
I know that it reduces strobe effect, but I'm not sure what is actually happening or how a camera can operate without a shutter?
The shutter on the camera opens and closes. If a camera has its shutter set to 1/100, each frame will be exposed for 1/100 second. But, remember that frame rate and shutter speed are independent. If you leave the shutter off, it will shoot at the frame rate.
There are actually 2 shutters: mechanical (which we all know) and electronic. The electronic shutter is how video is recorded. The sensor scans line by line either vertically or horizontally using the "rolling shutter" (you can read about it more in depth later) method at for example 25 frames/sec. It's recommended to use an actual camcorder to record video because the DSLR sensors will get hot after a long time.
Hope this helps.