Step 1: Open any flash game. I will be using Kass Basher for this example.
Step 2: Right click within the area of the container of the flash game, like this:
Step 3: Press "Inspect Element". You should see a huge bunch of code that jumps straight to <div style="text-align: center...
Step 4: Search for the line (highlighted in the image) that goes roughly
swf.addVariable('g', 'THIS IS THE MAGIC PHRASE');
Step 5: You want to look for that magic phrase that starts g_... with a bunch of numbers and letters after it. Now, type in this url in your browser:
images.neopets.com/games/[THE MAGIC PHRASE].swf
Step 6: On Chrome, just entering the url will automatically download the .swf file, like this:
On Firefox, it will actually load up the game on the tab. You will need to do File -> Save As to download the .swf file:
Congrats! You downloaded the .swf file!