You CAN'T do this with appendTo(). You can't do it by simply shuffling the order or nodes. And it isn't necessary to shuffle the order of nodes anyway.
The basic technique is to use a position:relative base node and position:absolute items, then manipulate CSS of the items. You need to position one over the other if you want to fade, or in starting position for movement effects, and then apply CSS transitions.
The plugin you found does this, supports a wide variety of effects and options, and seems to work well. Is there some reason you don't want to use it? Did you try it?
It is a bit old, and hasn't been updated in 3 years. There are certainly other similar plugins that are currently maintained. I would suggest you search for a newer/more maintained plugin. I'll bet you find a dozen.
A more modern plugin might utilize CSS flexbox to create the basic grid layout at load time, rather than JS doing pixel calculationsYou might have to be a bit creative to come up with search terms - I tried "brady bunch effect" but I didn't have much luck with that! ;) Figure out what designers call this... try e.g. "random image shuffle grid js css plugin"
If you want to do it yourself as an academic exercise, you can look at the source code of the plugin, and use it to make your own plugin more suited to your own needs, or just get some ideas from it.