We booked hostels to stay in and booked flights for the early morning (typically we found 5am or 6am flights). On the day we had to check out of our rooms, we would request to leave our bags at the hotel/hostel for the day (most places graciously allow you to do this), spent the day out and about being tourists, and then picked up our bags in the evening and headed to the airport. Some airports were nicer and more accommodating than others, meaning the seats were cushioned or there were a row of seats with no arm rests so we could lie down. We would use our bags as pillows (which was a good way to prevent theft) and we used the blankets we had swiped from our initial flight from the U.S. to Europe to keep ourselves warm.
I would never have done this if I were travelling alone though. It's always nice to have someone who can babysit your bags while you go to the toilet or look for somewhere to buy food.
It's also a good idea to scope an airport's website beforehand to make sure there is seating (because some airports don't have places to sit except in the areas you can access after you check in). Bring a sleep mask if you're the type of person that needs darkness to be able to sleep (since you can't exactly dim the lights in the airport) and it's also good to have headphones to drown out the noise of the cleaning crew.
It was actually a pretty nice deal. It wasn't horribly uncomfortable and we saved a bunch of money!