There are a multitude of reasons why people stammer; these are the 3 main ones:
Developmental - Most common in children younger than 5 years old, particularly males, this type occurs as they develop their speech and language abilities. At this time when children start to develop their speech it takes a lot of communication between the different areas of the brain and the muscles that are tasked with talking and breathing. When all works in synchronisation then speech can be without a pause because everything is working together at the same time and at the same pace to allow for rhythm in speech.
Neurogenic - Signal abnormalities between the brain and nerves or muscles cause this type. This is where the areas of the brain aren’t working in synchronisation and when this happens the nerve connections can break down causing interruptions in speech where the messages haven’t been transferred from the brain to nerves and muscles either at all or in the correct order.
Psychogenic - This type originates in the part of the brain that governs thinking and reasoning.
There is also a link to genetics, as roughly 60% of people who stammer also have a close relative that stammers. This points to genetics playing a role, however it is unlikely that a stammer is purely genetic.