Grammatically, you avoid two negatives in a sentence because they cancel each other out. Literally, "a little party never killed nobody" means "a little party always killed somebody". However, in this case, your understanding is right: "a little party never killed anybody" is the "proper" meaning. Pop songs tend to break rules and use street slang because that's what the kids like.

In spoken English, even native speakers violate the "prescriptive rules" of proper grammar. A strict teacher will correct you, requiring you to say "a little party(ing) never killed anyone/anybody." However, in spoken English, people will use the double negative such as the song title you cited or "I ain't seen notin' (nothing)!" But note that doing so will perhaps unfairly mark you as being uneducated at worst, and "folksy" at best.