Why does the family open the door to admit the prophet Elijah?
The answer to your question is actually fairly simple and steeped in Jewish culture (especially in reference to the Passover meal).
It is a part of the Passover ritual to have the family open the door and admit the prophet Elijah from the Old Testament. It is generally a ceremonial opening in that no one expects the true prophet Elijah, back from the dead, to actually enter the household. Instead it is simply a welcoming of the prophet's promise.
In The Devil's Arithmetic, it is Hannah who is asked to open the door on this special Passover day. However, instead of it being the usual ceremonial opening, it becomes much more. Hannah expects to see the apartment hall in front of her (as she has every day of her life), but instead she sees a field of green grass at night.
In conclusion, Hannah has done the Passover ritual of opening the door for the prophet Elijah, and she is a bit afraid, but simply figures this must be a game or a dream of some sort. Soon Hannah meets her uncle Shmuel, and her real adventures begin.