Answer:
Most games on Addicting Games are Created with Adobe Flash.
There are many open source (free) ways to create Flash Games (like FlashDevelop - http://flashdevelop.org/), and there are more costly ways that generally require less scripting (Flash Professional - http://adobe.com/products/flash).
I personally prefer Flash Professional, and it is by far the best application for creating Flash games.
You can find more open source applications for creating Flash Games at http://osflash.org/