Frame stacking effect in After Effects
Think about what you're asking. When you did the horse you used a luma key to isolate just the horse. If the horse was running through trees or down a racetrack what would you want to happen to the trees and other stuff in the background? Would they go over the horse from the layer underneath? If so, how are you going to see the layer underneath?

There's no way of overlaying frames without the front layer being partially transparent - either transparent in part of the frame (masks or roto), transparent depending on their pixel values (transfer mode, or key) or with an overall opacity of less than 100%. If it's not transparent then you can't see the background, it's that simple.

If you can, shoot against a green screen and key out your foreground. This will let you layer it up to create a visual echo as you show in your example. That's the best way to do this, even though it's "cheating".

If chroma keying is not possible, and if only part of your frame is moving, you might be able to use a difference matte. Get an empty shot of your background and use that to matte out only the parts of the frame that have changed. This will work best with locked-off shots.

Failing that, have a play with various transfer modes. They're basically pixel level keys that work depending on the value of the foreground and background colour. I worked with a visual artist who produced some lovely dance videos using time-delayed layering kind of technique. She filmed the dancers against black, and rather than pulling a key we just used the lighten transfer mode - which is basically a way of keying parts of your image that are darker than the layers below (we were using hundred of layers in each comp, I did some scripting work to make the whole process a lot easier).

Otherwise, if transfer modes, chroma / luma keys or difference keys don't work, put on a pot of tea, turn up the music and get stuck in to the roto.