There are lots of reasons. Some off of the top of my head:

1. People are more likely to shop there because they get “deals”.
2. People feel better about shopping there because they get “deals”.
3. More and better data for the business. (Associated with individuals over time rather than “anonymous” purchases, the also get extra info like a phone number that they can cross-reference)
4. If you carry the card or app you will see it frequently and think about the store (free advertising).
5. Often times you agree to some sort of marketing communication when you sign up.
6. You usually get “points” which you need to come back again to use.