On the page you want to add the video to, click "Edit." Once at the editing page, you'll see this:
Click the icon I boxed in red. It will say "Add / edit embedded media."
When that window pops up, copy and paste the URL from YouTube (from the address bar) into the space that says "File/URL." Then ONLY CLICK in the "Dimensions" box -- YouTube videos have an automatic dimension, so all you need to do is click. Then click insert.
After you click "Insert" your video will appear like this:
.. But that's only on the editing page. After you click "Publish your changes" it will come up normal, like this: