"They need to read that book again."
Translation:Họ cần đọc lại quyển sách đó.
I think when the lại comes after the verb it shows that exactly that action is being repeated. In this case they read the exact same book again / start from the beginning/start all over / redo.
Other example: I write an assay, then pour coffee over the pages. So I have to repeat the action of writing the exact same assay again. So the lai comes after the verb. I have to start over again.
When the lai comes before the verb it means that an action like that is picked up again. Eg. I handed in my exam and i already have to write the next one (not the same!) then i have to "lai viet" it again. The lai comes before the verb. Other example: I haven't read in a long time and now start reading books in general again (also books I have not read yet) - then the lai comes before the verb.
Dunno if that makes any sense, I find it hard to explain this ^^'