What should I write about the reasons I wish to attend the Summer Intensive Course? Should I write an essay/covering letter?

You should explain what is your particular interest in the Greek language, especially when this in not obvious from your field of studies. You should describe how you would benefit from this Course and in what way this program will enhance your academic achievement record with regards to your background C.V. If you wish to back your application with an essay or covering letter you are welcome to do so.