What is the meaning of the words Après le déluge?

The meaning is 'After the flood'.

The words Après moi, le déluge -- after me (comes) the flood -- are attributed, as a premonition of the French Revolution, to King Louis XV of France (1710-74), although if anyone in fact used the phrase or one like it in this sense, it is more likely to have been his mistress Madame de Pompadour (1721-64), who is said to have predicted Après nous, le déluge : after us (comes) the flood.

The widespread use of that expression in French would be best translated as "after me, comes what may" in the sense that you don't bother.