When we compose our images through a viewfinder, it is easier for us to ignore things outside the frame and pay closer attention to what lies within it. We can then make conscious decisions about what we include and crop out, which ultimately produces stronger compositions.
When we capture an image through our LCD screen we are tempted to immediately check it over for flaws and reshoot if necessary. While this probably sounds like a good thing, it can result in us becoming a little lazy, knowing that if something goes wrong we can just take another photo. By taking a film-like approach to our photography and ignoring the LCD screen for a while, we can improve our skills and chances of getting the right image on our first try.
Our last point on the importance of viewfinders relates to camera shake. When holding our camera up to our eye to take a shot, we are able to brace ourselves a little better and hold the camera firmly, which eliminates some of the small movements that lead to blurry images.