Because if you put 'for' instead of 'efter' in the sentence you would get "Jeg læser det for hende" which means that you are reading it aloud for her. By putting 'efter' you get the sentence "Jeg læser det efter hende" which means that after she is done reading it you will then read it.
Ahhh. It took me a while for me to figure out what you were saying. What had me confused was that the first translation "I read (reed) it after her" doesn't make grammatical sense so I read it as "I read (read) it after her" which does make grammatical sense. It, however, also changes the meaning of the sentence.
It makes grammatical sense though we would use it with more words. "Every week I read it after her." (reed) implies a regular occurrence. We just don't know the context. We could have been talking about how someone is improving their reading skills with a tutor. The "it" could be a book that they are reading a chapter from every week. Reading aloud is a skill that is useful in school. Early readers can benefit from taking turns reading paragraphs.
Without further context, it is incorrect. Past tense or will-future are OK in this sentence. Simple present is not. Unless you stipulate that it is eg. a magazine that -every month- you habitually read after she reads it.
As it is, this kind of construction just teaches non-English-natives bad English.