What exactly do they mean by skin hardening, thickening, and shiny?

Exactly what it says. If your skin is losing its normal pliability and feels "hard" under your fingers, it's "hardening". If it feels thicker than normal skin, it's "thickening". And if it reflects more light than normal, it's "shiny". Those changes are often used to describe scleroderma and its clinical variants, especially when accompanied by stiff joints & cold fingers/toes.