Dativ Plural - mit den Leuten Hier eine Uebersicht: http://www.udoklinger.de/Deutsch/Grammatik/DekNomen.htm

Ja, Franziska is right ! so , "mit Leuten" means " with people" - it si Dativ, that's why you need "N" . for example "Mit Freunden" means "With Friends" , so you must always ask at first " with whom" and this question needs Dativ. And when Dativ all nouns need "N", of course when they don't end on "N". The word like "Eltern" (Parents) doesn't need "N".

Leute is plural already, there´s no singular and you´re right, the "n" is because of the Dativ. You do add an "n", unless the noun ends on n of course: den Mädchen, for example.