I added Business Phone and a few other contact fields to my view of the Tasks folder, but the phone numbers for the contacts in my tasks don't appear in the view. What did I do wrong?
The only thing you did wrong was to have high expectations of Outlook. The program simply doesn't work the way you imagine it should. In particular, it doesn't function like a relational database that combines data from multiple tables into a single query.
Outlook lets you drag any kind of built-in field to a folder view, but the view still shows only the data for that particular folder. The Business Phone data exists in the contact's record in the Contacts folder, not in the task in the Tasks folder that you have linked to that contact. Therefore, the field in the Tasks folder view will always be blank. You can get to the contact's phone number, though, by opening the task and double-clicking the underlined name in the Contacts box at the bottom of the task window.