More interesting than watching replays of my own raids, I want to see people raiding my Outpost! Then you'd know how you can improve it and the reactions people have would be priceless!
You can do that on the green, sometimes red, monitor to the left of the table. It has two categories, one for your raids, and one for people who raid your outposts.
There are someplayers I can watch the replay for, and some I cannot. I don't know what the distinction is (maybe they opted out of being used for replays?) but I have experienced both.
Oh, I don't know how I didn't notice that the replay menu is both your replays and people reading your base. Awesome!