We ask for phone numbers because that's how WhatsApp identifies WhatsApp users, making it easier for you to message friends and family quickly.
Please note that WhatsApp regularly looks at the phone numbers in your phone’s address book and then checks to see which of those numbers are verified in WhatsApp. Any WhatsApp users from your address book will appear as contacts you can message in WhatsApp. During this entire process, phone numbers are sent to WhatsApp for lookup, securely, over an encrypted connection. In order to know who you're chatting with, the app displays the names from your address book.