What causes sinus pain and congestion?
Inflamed sinuses are often associated with some kind of illness or irritant such as flu, the common cold, or hay-fever. As an example the cold causes sinus pain and inflammation because the virus is attacking/located in the nasal passages which causes swelling in the mucus membrane (the mucus membrane lines the sinus cavities). The swelling, along with increased mucus production, combines to create clogging and the pressure associated causes pain. Humidity will help loosen the mucus and anti-inflammatory medication should help reduce pain by reducing the mucus build up and swelling - thus relieving pressure. 

The increase in mucus production is because of the role it plays in our bodies - it is produced to trap foreign bodies and help keep our system clean. The more mucus we produce the more our bodies are trying to clean up. This can cause an excess which is difficult to clear such that our sinuses become blocked and dried mucus becomes an irritant.

Sinusitis is the painful condition of swelling in the mucus membrane. Sinus comes from the latin for bent or curved surface, and -itis is from greek meaning swelling or inflammation linked to disease.

In summary: Painful sinuses, aka sinusitis, are painful because of infection in the mucus membrane producing swelling and excessive mucus build-up.