I'll try to explain:
If you view a youtube video, do you see the button that says "embed"? Well if you click on it, you will get a code per se; take this embedded code from a youtube video for instance:
<iframe title="YouTube video player" class="youtube-player" type="text/html" width="480" height="390" src="https://www.youtube.com/embed/YDAPGLO-r4s" frameborder="0" allowFullScreen></iframe>
Now, take a look at the part I highlighted:
<iframe title="YouTube video player" class="youtube-player" type="text/html" width="480" height="390" src="https://www.youtube.com/embed/[S-HIGHLIGHT]YDAPGLO-r4s[/S-HIGHLIGHT]" frameborder="0" allowFullScreen></iframe>
That's the embedded code you need to put for the vid in your profile.
Since in this case, the code happens to be YDAPGLO-r4s, in the part where it says "youtube URL" in "edit your details" you type "v/" in front of it, so it'll look like this:
v/YDAPGLO-r4s
EDIT: Ninja'd, but yeah. :3