How can I embed the YouTube field module in a specific page?
Fields are designed to be attached to pieces of content (entities), especially the main content entities (nodes). Using youtube field, would mean you can attach a different youtube video to every type of content. This likely results in a textfield on every node edit page (of that certain content type) where you enter a URL or something.

If that is what you want, proceed happily. If you go to the manage display section of your content type you can alter the teaser view mode (switch in the right top) and remove the field from display so it does not show on the front page.

If that is not what you want and you only want one block to appear in a sidebar or something, go to the blocks page, create a new block and just paste the embed code there. Make sure to set the content filter to full-HTML so it does not get sanitized. Each block allows you to specify the URL's on which it should be displayed. (There is no need for this module)