I would like to write an Applescript and attach it to a folder (I guess that would be called a "watched folder") such that the Applescript would trigger the running of a Filemaker script upon the arrival of a new file into the watched folder.

Is there a way to do this? I am lost in the Applescript syntax.

You would control click on the folder and pick attach folder action. it will become a watched folder as long as it is written like a drag and drop script. Let me know if you need more information, I have written scripts like this and I can give you examples.