To place a you tube video into your page, you need to copy and paste the
code provided in the you tube video page/s to embed a video in your own web
page.
You can't get the actual file to host locally on your own host's equipment.
Your host may not allow streaming video (which means you see the video as it
is downloading, rather than waiting for it to fully download before it
starts playing).
Your visitors would need decent broadband connection for 'streaming video'
(or even 'normal' video since these can be quite large in file size (in
terms of megabytes/gigabytes), depending on if you have high quality or not,
and there are other factors.
The You Tube embed method is neat and seamless.
http://andrews-homepage.net/?page_id=102 is an example of an embedded video
in a Wordpress blog, but the idea is the same, you just copy and paste the
code.