We ask for phone numbers because that's how WhatsApp identifies WhatsApp users, making it easier for you to message friends and family quickly.
WhatsApp uses the phone numbers from your phone's address book to provide you an up-to-date list of WhatsApp users you know to make it easier for you to message them via WhatsApp. WhatsApp regularly looks at the phone numbers in your address book and then checks to see which of those numbers are verified in WhatsApp. During this entire process, phone numbers are sent to WhatsApp for lookup, securely, over an encrypted connection. So that you know who you are chatting with, the app then displays the names from your address book.