In this anime, some certain magical doors appear wherever they pleased and let in the people of fantasy world to the restaurant in our side. But the title's bugging me. Why do they have to put 'to' instead of 'in'? If I put 'in' there, would it be awkward? Could you give me some examples that have the same usage of that 'to'?
There is a distinction between a door (restaurant, opening or similar) to another world and one in another world.
The preposition to indicates that the place or object concerned serves to transport a person from one world to another. It's like a bridge. The visitor enters from a particular world and is transported metaphorically into a different world - one that is more beautiful or strange or mysterious or whatever.
In this sense, diving equipment transports a diver from a land world to a sea world by allowing the diver to explore the world under the water.
The preposition in indicates that the door is already to be found in a different world and allows a visitor to enter (or leave) the venue concerned in that particular world - but not to leave the world itself.
So I might say that:
Explorers found a hidden door to the tomb of a pharaoh
indicating that the door enabled them to enter the tomb of the pharaoh from a passage or point outside the tomb.
Explorers found a hidden door in the tomb of the pharaoh
indicates that they were already inside the tomb when they discovered a door that led elsewhere.
Thus in your example a door to another world suggests that the restaurant offers diners an experience that is totally new to them. A door in another world would suggest that they were already in the other world when they came across a door that led elsewhere within this world. This is not the idea that the restaurateurs wish to convey.
I'd use "in":
Where is the restaurant? In another word.
But the door is "to" or "into" the restaurant in another world.
Perhaps, they take the restaurant as a kind of road which leads somewhere... (to another world). That's the only explanation I can give. Note that this use of "to" after "restaurant" is definitely non-standard because usually restaurants don't perform the function of a door or a road.