My impression (others can tell me if I'm right or not) has always been stamina indicates how long it takes a player's level of fatigue to increase.
So if you had two players, one is 99 stamina and one is 70 stamina, the first one will still be fresh at the point the second one is only fairly fresh or getting tired (depending on other factors).
It doesn't mean the 99 guy will perform any better when he does eventually become fairly fresh or getting tired or whatever, but it DOES mean he'll take longer to get to those lower levels (and perhaps recover faster too).
By the way, with most of my teams I set everyone to fairly fresh unless I have a reason not to do so because I want them all performing at their maximum (and usually starters end up getting more minutes anyway so it works out). Unless you have a star player who will still be better when tired than a backup would be at maximum, there isn't any reason to play them tired in my opinion.