Creating a web app in django that creates a video (with moviepy) and plays it: Where to start?
As I see, Django is for complex sites, you should look for Flask instead.
(I am learning Django and I know nothing about Flask, so I`ll go with it)
Here is the needed setup:

define the urlpatterns for handling the URLs
create a model for storing your video
create a django template for your page (html)
define a view for rendering out the template (passing the video)
maybe some css to design it

You can run your video generator(in the view) at every reload and override existing video(in this case you don't even need a model) or you can save the generated videos and capture the IDs in the URLs (for example: https://yoursite.com/1), in this case, the videos remain shareable.
If you go with the first option and sharing videos is not important for you then you could write a simple html page with a video and a button. The button can trigger a javascript function to run the video generator python script and refresh the page(the video is overridden), you may need to wait to the script otherwise the old video can load.
Restful API is a more advanced way to refresh the video, without reloading the page.
After these, you can deploy your page using an Apache server for example.
I tried to give you some guidelines (I am learning this on my own)
Hope that it helped :) There are tutorials for these.
I would like to make a web application where a user can:

click a button on a webpage to create a **random video (with moviepy)
play that video back on the webpage
click the button again to create a new **random video that replaces the existing one

** The random video will be created with a python/moviepy script that downloads a bunch of random video clips from the internet to a directory on my computer. It then compiles them into one video (.mp4 file). 
I've done the python script bit already which successfully creates the video file. 
To make the web app bit I have been recommended django and that's where I'm stuck! 
So far I have installed django and got to grips with the basics .. I have a home page that says "Hello world". 
My question is where do I go from here? 
How do I connect my python/moviepy script with django?
What steps, apps, components etc, within django should I look into to make this thing?
I'm new to coding and looking for some guidance. 
Thanks!