What do the file permissions have to be?

Depending on your host, there is a thing called UMASK. That is, the default permissions set to the file. Permissions in Unix land are a three digit number. 777, is all the way open, writeable and executable by everyone. 666, is everyone WRITEABLE which is what the script needs. Scripts, to the webserver runs as the user "nobody" so when the php script goes to write to the file, the system tries to write to the file as the user "nobody". Keep in mind, this doesnt mean anyone on the web can do it (unless through flyspeck script) just anyone logged into your particular part of your web space.

Any ftp client, I use CoreFTP a nice free one, allows you to right click the file and CHMOD the file something else. Some hosts have a control panel where you can also do this. I always try it at all defaults, and see if it works. If you get an error message, try chmodding the file to 666, most configs that will work. You can try higher settings and get the min one that works, or just go to 777.

Keep in mind, this is nothing particular to this editing script. Any script that allows people to edit files will have this same issue.