Why does Canon 600D/T3i video quality drop dramatically once it's recorded — in comparison to what I see in Live-View w/ Digital Zoom?
I'm not sure what quality drop in the video you're referring to, but the video quality will never be as good as the image you see in the Live View of your camera due to the way it works.

Here's how Canon describes the Live View mode:


  When the camera is in [Live View] mode the reflex mirror is in the  up
  position, blocking the normal pentaprism viewfinder.  Instead, the
  image is projected through the lens and  directly onto the sensor,
  which in turn displays the image  on the rear LCD panel.


Source

So the Live View basically shows you the image that reaches the lense, taking focus and depth-of-field into account. It isn't actually a preview of your video recording quality.

Here's what you can capture with your camera:


RAW pictures. While normal low- to mid-end cameras usually only capture compressed images such as .jpg-files, the RAW format contains all information captured by the lens. This is why you can fully adjust settings such as exposure, white balance, temperature and tint in post-editing. No compression is happening between the information captured by the camera (lens) and the saved file.
Compressed pictures such as .jpg-files. The information captured by the camera is compressed into a pixel-based image using your current white balance and exposure setting to calculate the pixels color values. Canon cameras can generate compressed pixel-images with a high pixel count (for example, the maximum resolution of the Canon 600D camera is 5184x3456, about 18 megapixels). So there is some compression.
Video. Video capture quality is limited by the cameras image processing speed and, to a lesser extent, by the writing speed of your SD card. The camera can't possibly capture about 25 images per second (assuming 25fps) in a resolution as high as that of single images. The maximum video resolution of most Canon DSLR-cameras is Full HD, i.e. 1920x1080. So you have a much lower pixel-resolution, therefore there is much compression.


The Live View shows you the information captured by the lens on the (obviously pixel-based) LED monitor, so its quality is somewhere between the RAW-and JPG-qualiy. This is why you notice a quality difference between your Live View and the captured video.

There are professional video cameras that can capture RAW-video (e.g. Alexa cameras which capture ARRI-video), but they cost several thousand euro.

How to get rid of it

If you are seeing grain and artifacts because you're filming in a low-light situation, use better lighting. If you want a dark scene, do it in post-production.