Most people use the system on a smart phone, which is a more personal device than a computer. So it makes sense to leave users logged in for extended periods for convenience. But some people find themselves having to use a shared computer sometimes, perhaps in the flying school. That's why this option exists:
"I'm using a public computer" tells the system not to leave you logged in if you re-open the browser. Technically, it changes the cookie setting from a persistent cookie with a lifetime of 23 days to a session cookie, good until the browser is closed.
But the problem with session cookies is they keep you logged in for as long as you leave the browser open. You still have to remember to click Log out, close the browser or turn off the computer. If you don't, it means the next person using the computer will have access to your account. If you're an instructor that means they can export the whole user list, all your bookings, raise invoices and so on.
From now on, if you've logged in on a public computer you'll need to use the system every ten minutes to stay logged in. That's much better security but it might affect you if you get distracted while in the middle of filling in student notes and you're using a shared computer.