Is it appropriate for me to connect with him and send a message in addition to my application, so that I stand out? What should I be trying to say or prove in that message?
Don't know where you heard it was a good idea, but I would suggest you don't send such message, at least not with those intentions and content.
Surely, there is no problem in you connecting with them, as it is quite common to connect with people you don't necessarily work for/with but are part of the industry and area of knowledge you are.
However, sending such message "to stand out" could be perceived as you trying to bias their decision process in your favor. This could be frowned upon and affect your chances of landing that job.
In any case just connect with them and send a friendly and professional message (or just connect), but I would suggest you don't try to bias their processes in any way; your application and qualification should speak for yourself, without having to recur to lobbying techniques.