Compliments of the chef [closed]

I'm a native English speaker and I've known of the phrase

"Compliments of the chef."

my whole life, but I've just realised I don't know what it means. With a search I found meanings for things like:

(As a diner) Give my compliments to the chef. (Meaning the diners enjoyed the cooking).

But "Compliments 'of' the chef" means something different. Could someone explain what this means? I know what a "compliment" is, is it that in this case the chef is "complimenting" the diner? If so how? And could you provide a link to an explanation?

Thank you.

Solution 1:

OED has

3. A complimentary gift, a present, gratuity. to make one a compliment of (a thing).

This is directly related, in fact, to the meaning that you cite in "give my compliments to the chef" — in either case, a "compliment" is a token of respect. But in practical terms, in a modern restaurant, when you receive an item "compliments of the chef", it's meant to tell you that you won't be expected to pay for it, and there's no need for you to say something like "I didn't ask for this, please take it back!". "On the house" is a synonym.