It means that difficult times will only be with us for a short while --- maybe for a day or a week, for example. When things "last", they are with us for a long time. Tough people, however, do "last" --- they keep on living for a long time after the tough times (the problems) have gone away.
Here are some sentences with the word LAST, so that you get the idea of other ways it can be used:
That's a huge bottle of soap --- it will probably last me a year!
Boy: Mom, how long will this boring meeting last?
Mom: It'll only last a little while longer, and then we can go home.
Andrew: How long can a goldfish last without food?
Audrey: Well, I've forgotten to feed him a couple times, so I know they can last a few days!
Hope this helps!