What does it mean when the unit of moles is raised to the power of -1?

Raising any number to the power of -1 is the same as 1, divided by the original number. For example, 4 to the power of -1 is the same as 1/4, or 0.25

Moles can be raised to the -1 power pretty frequently if you're doing stoichiometry or other forms of dimensional analysis. If you had 1 mole of something, and it had a molar mass of 5 grams, you could just simplify this whole thing into an equation like "1 mol x 5g x mol e-1." This is an alternative to writing out the "railroad track" method, where you write each relationship in the form of a numerator and denominator in a separate line. The mole raised to the -1 power would cancel out the first mole, leaving you with an answer in the unit of grams. 

Basically, any time you have a relationship between a unit and a mole, and the value is "N units per mole," you could write it as "N units x mol e-1" instead.