One of the options in the X menu should be labeled "Battle Video" (it's a dark blue color, and you might have to scroll over to page 2 to find it). Selecting that item will give you a curious one-item menu with a prompt to "Open the Vs. Recorder" and, upon selecting that, you'll see a list of whatever battle videos you have recorded.
Alternatively, you can go to Festival Plaza, enter the big castle, and just inside to the right you'll see a computer. Talking to that is another way you can open the Vs. Recorder (as long as you can connect to the Internet first), but the thing about accessing it from this interface is you can upload your videos by choosing the menu item "Make a Battle Video public" (but you can't have more than 10 videos made public at once), or you can enter battle video codes (each one a sequence of 16 letters and numbers) to view battles that other users have uploaded. Note that you can't watch a battle video that was recorded on the original Sun or Moon games, or anything on the Ultra games that predates the 1.1 patch. (Oddly, the update from 1.1 to 1.2 did not render previous battle videos incompatible, the way 1.0->1.1 and indeed most patches on previous games historically have.)
