Every Discussion and Page will have a rich text editor, whether you are starting a new thread or replying to someone within an existing thread. From the rich text editor, you'll want to click on the Apps icon that looks like an electrical plug:

Next, select My Media from the list of options. If you don't see My Media, select View All and locate it from the list that appears. This will bring up all of the content currently stored in your My Media:

Locate your video from your My Media library by typing the video title in the search bar or scrolling down. You can also click the +Add New button to upload a new media file, though you won't be able to leave the page until after it has finished uploading. 

Once you've found the video you want to embed, click on the blue </> Embed button to the right. The My Media menu will disappear and the video will appear in the rich text editor.

Once your video is embedded and you've finished adding any additional text/info to your post, click Post Reply or Save.