Alright, so let's start with the simplest part, hiding things. You can inversely use .fadeOut() the same way as you would use .fadeIn(). But that's assuming you want to add the effect as well as hiding it. jQuery also sports a .hide() function which will instantly hide an element, after which you could fade it back in.
So moving on, to display these fullscreen images you will likely want to create a new element on the page. Granted this could be done with jQuery itself but I think it's always easier and a lot safer to manually add a new <div> element to the page and just use that. Give it an ID and use CSS to have it cover the screen. You could use:
background: rgba(255, 255, 255, 0.8);
This will have the <div> cover your screen when visible, but it will be hidden to start. Then in each logic statement that determines the winner you would do something like:
$("#newImageDiv").html('<img src="/some/folder/rock_v_scissors.jpg" alt="" />').fadeIn(1000);
So then each time a game has been decided you should see your new <div> fade in, covering up the screen (and the play buttons) while displaying the appropriate image.
The last step would be using setTimeout() each time a game ends to show the replay button. Since this is a global action (not specific to the choices) it doesn't have to be inside of any decision logic. It would just call a .prepend()/.append() and .fadeIn() on that same new <div> to add a button that runs a function you will create. Your new function just has to use .hide() on your new <div> so users can play again.
This is a pretty rough outline and you'll probably want to clean up some of these things (like the large image. It wouldn't be vertically centered). I also didn't really address your results list that shows the text version of each result. I'm not sure what you want to do with that if you are adding images to represent this.