What does お任せ Omakase mean?

Omakase (お任せ) means leaving it up to you, in any situation. The English restaurant sense is a narrowing of an ordinary act of entrusting.

Written
お任せ (omakase)
Literally
From 任せる (makaseru), to entrust. With the honorific お, it means leaving the choice to the other person.
As slang
Ordinary and useful, not a culinary term. It works at a hairdresser, in a shop and in any decision handed to someone else.
In anime
Rarely about food. A character saying it is handing a judgement to somebody, and often the point is who they are willing to hand it to.

Omakase means I leave it to you. English has borrowed it as a restaurant term for a meal chosen by the chef, which is one true instance of a much broader everyday word.

The verb underneath

任せる (makaseru) is to entrust, to leave a matter in someone's hands. The polite prefix softens it into the form used when handing a decision over, and the whole expression is short because it is used constantly.

Where it is actually said

At a hairdresser, when the customer has no specific instruction. In a shop, when choosing between options the seller understands better. At work, when delegating. At any restaurant at all, not only expensive ones, when the diner would rather the person who knows decided.

It is a routine social move, and its content is an admission: you know more than I do about this, so you choose.

What the English narrowing lost

The trust. In English the word arrived attached to a particular kind of expensive meal and became a category of dining, so it now signals price and prestige rather than a relationship between two people.

The Japanese sense has no prestige in it. Entrusting a haircut is the same act as entrusting a menu, and neither is a luxury.

Why it fits the rest of this vocabulary

It belongs with yoroshiku and amae. All three arrange an interaction around one person relying on another's judgement or goodwill, and all three are ordinary rather than remarkable.

English can express the same thing and has to build a clause each time: you choose, I trust you, whatever you think best. What it does not have is a single set expression that hands over a decision without seeming either lazy or ingratiating, which is the slot this word occupies.

The restaurant sense, fairly

It is not wrong, and it did not appear from nowhere. A chef deciding the courses is a genuine instance of the general act, and English needed a name for that format.

The complaint is only that the borrowed sense is now the sole sense in English, so a word about trusting people has become a word about a price bracket.

Quick answers

What does omakase mean?

From 任せる (makaseru), to entrust. With the honorific お, it means leaving the choice to the other person.

Is omakase an insult?

Ordinary and useful, not a culinary term. It works at a hairdresser, in a shop and in any decision handed to someone else.

What does omakase mean in anime?

Rarely about food. A character saying it is handing a judgement to somebody, and often the point is who they are willing to hand it to.