Why do Jewish people open the door to let Elijah the prophet in?

After the festive meal at the Passover Seder, the door is opened while chanting verses asking God to redeem us from those that oppress us; this is done to show that it's the "Night.

He was the last prophet to hold the keys of the priesthood. His mission was to restore the keys of the priesthood, in order that all the ordinances may be attended to in righteousness.

Elijah shall reveal the covenants to seal the hearts of the fathers to the children, and the children to the fathers.

So what does this mean. It means that all will be responsible for gathering the names of all their ancestors so that vicarious work may be done for them by the living.

The scriptures talk about "...if the dead rise not at all why are the dead baptized? (Paraphrased)

The door is opened for the prophet Eliyahu because there is a prophecy that he will return to earth before hamoshiach arrives to anoint him. We open the door and have a glass of wine poured so that if he were to return, he would be welcomed.