Translation:She wants to see her children before they grow up.
What Jena said. Though about.com has a few more examples, and in my opinion, explains it better: http://french.about.com/od/grammar/a/negation_form_2.htm
The only expression in English like that is "Let's see if we can't." which means "Let's see if we can." You cannot arbitrarily put "can't" into other expressions. This ne is used for emphasis in subjunctive and is not a true negative which would have to have a second part, "pas" for example, in order to be negative. See the above website for explanation of this expletive.
In reply to the rodolfo.ornelas question of DL accepting "...before they grow old". Typically "grow old" refers to later on in life and "grow up" refers to earlier in life. One would assume that she wants to see her children before they become adults and not before they retire.
This particular sentence uses subjunctive because it expresses someone's desire which is either subjective or uncertain. The subjunctive is nearly always found in dependent clauses introduced by que or qui, and the subject of the dependent and main clauses are usually different. Click here for the whole story on subjunctive: