Reaching the play's final act, the audience/reader should understand that this is a play about the basic cycle of life. Pronouncements to this end are made at several points by the Stage Manager.
The last act is predictably concerned with death as the final basic element of the life cycle. As one of the central characters, if not the most prominent character, Emily is featured in the final act.
Her death then is not entirely surprising as 1) the play was leading to up to death thematically and 2) Emily is a natural choice for the character to be featured in this final act.