Mildly Infuriating

They can track what people are buying just as well without needing to know who is buying what. If all they cared about was making sure they are stocking items that are popular with customers, they could just monitor how many of the item were scanned at the checkout counters.

When you give your phone number to the store, it serves three functions:

1. The rewards scheme encourages loyalty. Once you have spent hundreds at the store, they give you a minor discount on something small, which saves you only a little bit of money. You only need to decide to patronise this store one time over their competitors for this bribe to be worth it.
2. By learning your shopping habits, they can offer you personalised advertisements to encourage you to buy certain products. For example, a grocery store chain which has a location near where I live has learned that I always buy a certain brand of cheese, and so occasionally they will try to tempt me with something like a 50 cent discount on that cheese, or they will send me a message saying "Hey do you still want (the cheese brand)? We have it in stock!"
3. Some people, when putting things into their basket or shopping cart, will see the large, advertised discounted price and think "this item is reasonably priced", but then they forget to put their phone number in when checking out and are thus charged the inflated price. The store pockets the difference as customers are unlikely to notice or complain about it. Most people do not closely monitor the price of items as they are being scanned. They only look at the total price at the end before tapping their card.

This helps them to manage their stock, keeping popular items that regular shoppers buy in stock etc.

Nope. Normal stock management works without tracking individual users habits.

These cards have the benefit of binding customers to your chain (and to sell ads).