What does 'Laissez les bon temps rouler' mean in English?

Let the good times roll!

This is English transposed into Cajun - an American derivative of French. Most purist French speakers wouldn't say it. The equivalent phrase in French would be something like 'que la fête commence', or 'allez, on va s'amuser, on fait la fête, qu'est-ce qu'on s'amuse!' and afterwards we would say 'qu'est-ce qu'on s'est bien amusés, c'était trop bien!'

(* also rendered as "Laissez les bon temps roulez", not technically grammatical as "rouler" is the infinitive form)

Actually, "laissez les bon temps rouler" is also ungrammatical, since "les" is plural and "bon" is singular. The phrase could be put in either singular or plural, but not in both at once! "Laissez les bons temps rouler" ("Let the good times roll!" [with "les" pronounced "lay"]), or "Laissez le bon temps rouler" (literally, "Let the good time roll!"--that is, "Let's have a good time!" [with "le" pronounced "luh"]).
The English translation is: let the good times roll.

As a phrase, "Laissez les bon temps rouler!" is the slogan for the Mardi Gras celebrations held annually in New Orleans, Louisiana. Mardi Gras (Fat Tuesday) is the final day of the festivities and always falls before Ash Wednesday (as determined by the Christian calendar, usually in either February or March).