Updated 3/26/2016: Microsoft is now rolling out the ability to download videos in a much more elegant manner.
One of the more popular features requested in Microsoft's Office 365 video portal is the ability to download the videos for offline use. As it stands today the user experience is that the video streams via your browser or mobile app and the user is not presented with any kind of download option in the UI.
But, we know that the Office 365 video portal is simply built on top of SharePoint so there must be a way to get at the actual video object somehow, right? Good news: you can, as long as you can construct the correct SharePoint URLs.
The root of your O365 Video portal is at this address:
Hitting that URL will simply take you to the O365 Video home page, which isn't super helpful for saving a video. In order to download a video you'll want to first identify the channel the video is in and then access this URL to see a SharePoint view of all content within that channel:
https://[Tenant Name].sharepoint.com/portals/[Channel Name]/pVid/Forms/Thumbnails.aspx
Note that the /hub part of the previous URL is not necessary here and if your channel names have a space remember to replace the space with a hyphen, not your typical %20 encoding in the URL. SharePoint replaces the spaces with hyphens when it builds the site URL.
As an example, if my tenant name is confusedamused.onmicrosoft.com and my videos are in a channel called PDX then the URL to see the contents of that channel would be this:
Once you land on that URL you should see thumbnails of all your videos. Hover over any of them and hit the ellipses button to present a download option. Be sure to watch your video at least 5 times to justify the extra effort required to save it for offline access.
Alternatively, if you know the channel and video name you can access it directly with this format:
https://[Tenant Name].sharepoint.com/portals/[Channel Name]/pVid/[Video Title].mp4
In the case where your video title has a space in the name you'll want to use a %20 encoding as usual. And one last note - if a video channel name was ever changed the SharePoint URL will retain the original channel name's URL path despite the new display name. You can sort out what the original name was via some Fiddler traces if you're striking out on the URL you've created, but that is outside the scope of this post.