Why does Holmes want Helen to be precise about the details of her story about her sister's death?

Sherlock Holmes always wants his clients to be precise as to the details of any problems or stories they tell him. But there is another reason for this and other interjections by the detective while he is listening to Helen Stoner's necessarily long back story. The author, Sir Arthur Conan Doyle, wanted to break up Helen's story by making it seem more like an interview or a conversation between Helen and Holmes, and not just a long story within a story. Without all the interjections, the reader might think that Helen's back story was the story, whereas it is only a prelude to the real story. Holmes has to say something to show that he is listening and thinking--even to remind the reader of the presence of both characters. Holmes makes many interjections for just that purpose while Helen is speaking. Here are some examples:

“Your sister is dead, then?”

“Pray be precise as to details,” said he.

“Perfectly so.” (In reply to Helen's question, "Do I make myself clear?"

“Indeed,” said Holmes. “Was it your custom always to lock yourselves in at night?”

“And why?”

“Quite so. Pray proceed with your statement.”

“One moment,” said Holmes, “are you sure about this whistle and metallic sound? Could you swear to it?”

“These are very deep waters,” said he; “pray go on with your narrative.”

And so on. These interjections are unnecessary, since Helen is telling a very precise and coherent story. They only serve to break up that rather cumbersome exposition of her presenting problem. Helen has to go back a long way in time and offer a lot of detail because the fact is that the real question is not about Helen's fears for her own safety but about what happened to her sister Julia two years ago.

Modern readers might not have the patience to read such long introductory material as that contained in "The Adventure of the Speckled Band" as well as in "The Red-Headed League" and in other stories. But Victorian readers had no media entertainment other than books of fiction, life was slower, and people were so fond of Sherlock Holmes that they enjoyed everything about him, including his speech, habits, idiosyncrasies, and his lodgings and furnishings at 221B Baker Street.

"The Adventure of the Speckled Band" is somewhat unusual because Holmes ends up saving his client's life by solving the murder of her sister Julia two years earlier. Julia was killed by Dr. Roylott to prevent her from marrying, and Helen is in danger of the same fate because she has recently become engaged. 

There is really no special reason for Holmes asking Helen to be precise as to the details about her sister's death. He is just showing that he is all attention and is breaking up her long back story with his frequent interjections. No doubt he will have all the details clear in his mind by the time the interview is over.