Developers estimate how long the stories might take to implement. Each story will get a 1, 2 or 3 week estimate in "ideal development time". This ideal development time is how long it would take to implement the story in code if there were no distractions, no other assignments, and you knew exactly what to do. Longer than 3 weeks means you need to break the story down further. Less than 1 week and you are at too detailed a level, combine some stories. About 80 user stories plus or minus 20 is a perfect number to create a release plan during release planning.
Another difference between stories and a requirements document is a focus on user needs. You should try to avoid details of specific technology, data base layout, and algorithms. You should try to keep stories focused on user needs and benefits as opposed to specifying GUI layouts.