Extremely dry hands can be caused by washing your hands too much, dry air, not staying hydrated, working with your hands to much, exposure to chemicals etc.
To keep your hands from drying out you should use some moisturizing lotion or Aloe Vera on your hands daily and avoid washing your hands too much.
Your hands and other parts of your bodies skin contain natural oils that keep your skin moisturized and when you wash your hands or body too often you strip your skin of those essential oils and that can then lead to dry skin and dry hands.
You can also moisturize your hands and other parts of your skin using some olive oil as olive oil is a natural moisturizing oil that helps to hydrate your skin.
Also make sure to drink plenty of water because not staying hydrated enough can also pull moisture out of your skin which causes your skin to dry and cause dry hands.