BobbyFK probably can explain it best and probably has a better method.
On laptop, here is what I do..
Find GIF you want to post. Note: BMR doesn't allow large GIFs or GIFs from certain sites. I use Giphy.com a lot so I'll use that as my example.
1. Click the GIF image you like out of the set of GIFs that come up when you do a search on Giphy.com. When the GIF opens in a new window, hit SHARE link and then select "Copy GIF Link".
2. Come back to BMR and start a new post. Select the 3 vertical dots in the top center of the reply post, just to the right of the "Insert Link" button (chain links looking thing). A second line of icons pops up. Hit the 2nd icon from the left (the Media button - the one next to the smiley face).
3. Paste the GIF link in the pop-up window and hit continue. It will not look like a GIF until you submit your post.
When you hit the Media button, you can see which media links are allowed to be posted.
Use same process if you are posting from YouTube, except instead of hitting the "share" button on Youtube, you just copy the Http address and past that into the media link.