On things made from cotton starch will lodge itself between textile fibers and form a surface coating. All of which smooths down normally fuzzy cotton threads and provides measure of soil/stain resistance. Things stick to the starch (within reason), and are thus washed away more easily at laundering.
That being said far less starching is done today than in days past. Even men's dress shirts once routinely starched to death now mostly are preferred without. Today's softer fabrics used for men's dress shirts and other items of clothing do not need nor should be heavily starched.
That being said a light dose of starch won't make things stiff, but will add a measure of soil resistance. Some men simply prefer to have collars and cuffs starched, but not body and sleeves of shirt. Certain collars look better with a bit of starch, and it does help with ring around the collar. Though washing one's neck is often a better thing.
There have been a couple of threads in this forum over years on starching. A quick search should turn them up.