The scenario editor is triggered (just like any scenario) from the 'Operations Selection' screen in the front end.

Once you have completed the 3rd level in the game (backlash) you can start the scenario editor which is called erm 'Scenario Editor'. You will find it in Africa.

Once started you will see the globe with a number of units and nations set up to begin editing your scenario with.

Once you are happy with your scenario you should:

Give it a name by pressing 'N'. This is the name it will show up under in Steam so make it good.

You will probably want to save a local copy using 'S' if you haven't already done so.

'W' will take you to the Steam workshop screen.

Here you can edit (D) the description users will see in Steam when browsing the workshop.

You can also set an optional 'Update message' to keep track of the changes made. This is less important.

Now you are ready to upload using 'U'.

It is now in the Steam workshop. You will now need to find it using steam client (or browser) in Steam. Here you may need to accept an agreement with Steam before it will go live.

Congratulations.
Your scenario is now available for the world to enjoy.

Use the steam client or a browser to look through the 'War, the Game' scenarios in the workshop. (I've done 'Tank Rush' and 'Cold War' but some of the others that are there look pretty sweet)

Still in Steam you 'subscribe' to the scenario. Steam will now always automatically download the latest version to your computer.

Now start up 'War, the Game'.
On the 'Operation Selection' screen start the 'Scenario Editor'.

Press 'L' for Load Scenario.

You will now see a list if scenarios available to you. Use the cursor keys to navigate the list.
The ones in green are on your local harddisk (you made them). You can Load these (RETURN) to continue editing or just play (P) them.
The ones' in orange are scenarios from the Steam workshop. You play these with RETURN or P.

This is easy:

Start the 'Scenario Editor' from the 'Operation Selection' screen.

Now right mouse click on a unit or city of the nation you would like to play.
Press 'A' to make that nation the player nation.

Right mouse click on any nation you want on your team.
Press 'T' to put that nation on your team.

Right mouse click on any nation you want on the opposing team.
Press 'O' to put that nation on the opposing team.

Press 'P' (3 times) to start the scenario.