You need to keep that box checked . It's all to do with cookies , which are small files the browser saves on your machine to remember things like settings on your computer.
Cookies have an expiry date/time after which they don't work though, otherwise you might log on to a shared computer at you local library say and two years later someone else log in to the same site and it would remember your logon details for them
There are two cookies involved here though. Note it's a long time since if worked of the internals of this sort of website and the times I mention here are from memory and also re-settable by the website administrator so may not be 100% accurate for this site.
The first is the one that remembers your logon details . It's expiry deadline is set according to whether the checkbox was checked when you logged into the website. If it was checked it sets it to expire in something like 1 years time, if not it sets it to expire in a short time like 30 minutes.
Now every time you open a new web page on this site it checks if there is a logon cookie set and it if has not yet expired it uses the details it finds there to check against your logon details stored in the website's database on the server. If they agree it shows you a logged on, gives you the privileges of a logged on user. It then resets expiry of the logon cookie as I talked about above, but based on the current time.
The net result of this is that if you logged on with the checkbox ticked you stay logged as long as you use the site once a year, but if it's not ticked, you will stay logged in as long as you keep using the site. If you pause however for more than 30 minutes you will be logged out.
The second cookie remembers the state of the checkbox. This also has a long expiry time.
However if you clear cookies, you not only clear the logon cookie, but the checkbox cookie too. Next time you you come to use the site you will be asked to logon again but the checkbox will also not be ticked. If you don't check it again manually you will only be logged in for the duration of that session, next time you will have to log in again.
Note that the same thing happens when the checkbox cookie expires - this is just to force you to logon once a year or so. This is mostly designed to protect people who sell computers without clearing the cookies ! Again if don't manually check the box then, you only get logged in that session and next time need to log on again.
To know the meaning of life you need not to be dealt a good hand, but rather to play the worthless hand you have been dealt to the best of your ability.