It is used here to express responsibility or requirement to do a particular action, in this case leaving at 2:32 pm. It means that she was in a position that she must leave by that time, but since must is only used in a future tense it could not be used here. If she was apologizing today for having left at that time yesterday, then it would be stated like this: She apologized profusely for having had to leave at 2:32 pm.
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