It's a quick fix. When you post a YouTube video in a New Post, you are copying and pasting the embed code available at YouTube into the "Edit Html" field. The dimensions in YouTube's embed codes are often pretty large, particularly with respect to the width of the video. For instance, the Uff! video just posted today is about 560 pixels wide, by default. I've estimated that the space allotted for Blogger posts is about 400 pixels wide. You can reduce the width of embedded YouTube videos by editing the embed code. With the Uff! video, the default code reads:
By reducing the dimensions of the video (the "height" and "width") in proportion to the space alloted by Blogger...
...you can ensure that the entire video appears in the Blog Post. I also use as a general rule of thumb that a height of 333 pixels and a width of 400 pixels works for most videos, though the dimensions may appear a little off, depending.
Hope everyone finds this useful!