First, allow me to further explain the phrase for you. "To let someone in on something" creates the mental image of a club. You can imagine that this club has a building, a clubhouse, for their members. The people inside the clubhouse understand a secret that only they know. You are asking to be allowed (let) inside the clubhouse, where everyone understands the secret. Thus, you are asking for an explanation of the secret.
Synonyms for the phrase become obvious at this point.
TV chef Raymond Blanc will explain the secret things he does in the kitchen that enables him to cook better than other people.
I see that everyone here is laughing, but I don't know why. Would you please explain the shared history that you all possess which makes the current situation funny?
Reveal your secrets.
Explain what I don't understand.
Teach me about this.
I don't "get it", but I want to (get it).