How does the varicella virus cause the symptoms of chickenpox?

It's quite a story! After contaminated respiratory droplets are inhaled, the virus infects the lining of the respiratory tract or of the eye. The virus multiplies in lymph nodes near the upper respiratory tract; next, the virus stays in the blood for a few days. Then it multiplies in internal organs; it returns to the blood and infects capillaries and skin cells, causing the classic blisters.
It is the cause. Chickenpox is caused by the varicella-zoster virus. The mechanisms of the disease pathogenesis are beyond the limits of this venue.