You don't need to ask for permission. I know it because I have my own videos there.
On the Right hand side of each video on youtube, just below the description, there are two boxes with some code. One says URL and the other one says Code to embed. You can copy and paste them anywhere you like, if they are there it means the video owner allows it, otherwise they could prevent people from embedding them.
CODE TO EMBED
<object width="425" height="344"><param name="movie" value="http://www.youtube.com/v/Vchy1ExvDws&hl=it&fs=1&"></param><param name="allowFullScreen" value="true"></param><param name="allowscriptaccess" value="always"></param><embed src="http://www.youtube.com/v/Vchy1ExvDws&hl=it&fs=1&" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="344"></embed></object>
When you embed videos they are on your page, visitors don't leave your website.
For video owners is a way to get more views, so most people allow embedding.
I hope it's useful