Why is the phrase "Emily's sister gave birth to a pair of twins?" redundant?
If 2 babies were born, then there are 2 ways we're being told (which is redundant). But if 4 babies were born, it isn't redundant.
Something that is redundant is something that is not needed - the information in the redundant part of the sentence has already been expressed in another part.
When we look at the sentence in question, we are told someone gave birth. Ok - so normally we ask "was it a boy or a girl?" and in this sentence we're told that there wasn't just 1 baby, but instead there was 2 babies. We're told this both by being told there are twins, but also by being told there are 2 twins. This is redundant.
It's like saying "I get up for school at 6am in the morning" - we know you get up in the morning because you've already said you get up at 6am, and vice versa. One of those bits of information is not needed and is, therefore, redundant.
As a side note, the sentence in our original question need not be redundant - it could be the case that 4 babies were born and that they are two sets of twins. It's not likely but stranger things have happened. (But then again, we'd probably say "E's sister gave birth to two sets of twins"... but it wouldn't be wrong to say a pair of twins when talking about 4 kids...)