Physically, why does a nose get clogged?

Swollen membranes: The most common reason is swollen membranes from allergy, cold virus, or irritants. Irritants are important to consider if you live in a city or are exposed to pollutants in your job. Of course, inserting objects in your nose can cause irritation as well.

Many possible reason: In terms of physical obstruction, there can be blockage from any of the anatomic components of the nose. Some of the more common components would include the septum or turbinates. Weakness of the nasal tissues can also lead to collapse on inspiration causing clogging. Missing/removed anatomy can cause airflow disturbances that can lead to the feeling of plugging as well.