Google added a new Blogger in Draft feature that makes adding the YouTube video a lot easier. You just click Insert a Video button in the toolbar. In addition to already existing Upload tab, you'll now see two new tabs: From YouTube and My YouTube videos. In the From YouTube tab, you can search, browse, and play back YouTube videos all within the interface. If you've your own videos on YouTube, My YouTube videos tab will show them as well.
Once you choose the video, click on Select to insert the video to the post. You'll then see a thumbnail image for the video, which you can move around in the post by drag-and-drop.
Also, note that, you can add a YouTube video to your blog while watching a YouTube video, just click "Share" button, and click on Blogger, and you'll see the familiar BlogThis! mini post editor -- all you've to do is to select a blog (if you've more than one blog, that is), fill out some content, and then Publish the post.