In early primary, many of the more "academic" skills you might hope to see are still a little ahead of many students' abilities. Having good fine motor skills means it is easier for a child to hold a pencil correctly to write, manipulate items to count, even hold books and turn pages. What you might see in the classroom instead are children tearing strips of paper, scrunching, stapling and stuffing balls into a shape, following instructions and describing to another child what to do.
The teacher will need to gauge how ready her class is for a lot of writing. Pushing a small child to hold a pencil for long periods of time to write a full page of text is exhausting and counterproductive if they can't hold a pencil correctly. Also, thinking about the story they are working on, the sentences to craft and which words to use decreases their pleasure in what they're learning.
Looking at a child's gross (big muscle groups) and fine motor skills will help the teacher decide on how to divide classroom time between muscle development and actual writing.
Farther down the arm are the bigger hand muscles used in scrunching, stapling and stuffing. If you have ever done this all day, you know how tired your hands will be. For small children it can be tiring after an hour or less. Combining activities like tearing, scrunching and stuffing gives a child the chance to rest one set of muscles while they use the other. The opposable thumb is something humans have over other animals, but it is still a muscle to be strengthened. Teachers are aware of the impact a lack of strength and coordination will have on their students' ability to fully engage in and learn from other activities.
Folding paper into origami or paper springs gives the finger muscles and the brain a workout. Think about how the brain has to see the item they are working on, communicate with the fingers and rotate the item to create the desired outcome. All of this is developing your child's abilities both to think, observe and physically manipulate. These skills will be so important in many academic areas later.
