There are 2 train stations near Cortona: Terentola and Camucia.
The earliest train leaves the first at 5:20am, the second at 5:25am and heads toward Arezzo, where you need to change trains to continue on to Florence's SMN train station. It arrives at 6:45am at SMN. From there, you'll need to catch a taxi to the airport if you need to be there by around that time, that is the fastest way to make it.... you can see details on
www.trenitalia.com, the official site for train timetables.
There are no trains to the airport, the tracks don't go there.
If, on the other hand, you mean your flight leaves at 6:45am, then the best solution would be to spend the overnight before in Florence and that way all you need to do is catch a taxi to the airport to be there earlier for check-in! Definitely much more enjoyable and stress-free, which is what a vacation should be ;-)