They can gather a lot of information about you that way. Information is $$$

Because if you have a Kroger’s card you’re going to want to shop there more often to get the free stuff. You’re not paranoid, it is a psychological tactic to get you to spend more money. Casinos do similar things.

They get even more detailed information about your buying and spending habits. They can use this information to better manage their stock and sales to maximize profit, and probably more importantly, they can sell your contact information and buying habits to other companies who are in the very lucrative business of aggregating consumer data and monetizing it.

First, more business. (If you get “rewards” like discounts or whatever, you’re psychologically more likely to return more often. Or at least some customers are and it’s not like they can really offer rewards only to the ones who are more likely to return.)

Second, and probably more significantly, data about you that they can use to do targeted advertising at you or sell to others (for a profit) who want to do targeted advertising to you.

If the rewards program involves a phone app or web app, that can get access to a lot of information on your phone or information about your browsing habits. And that can also be directly used or sold to someone else who wants data on you.

You’re definitely justified in being paranoid. It’s very much the kind of thing where if you stop buying Oreos regularly, Facebook will start sending you Oreo advertisements. (Oreos are just an example. It could be any company/brand that they advertise to you this way. But you know what I mean.)

Free stuff gets you in the door where you’ll be likely to spend money on other stuff.

Building ‘points’ or whatever to get free stuff gives you an incentive to buy things from Kroger you might get elsewhere. Ton of benefits to ‘gamification’ of damn near anything.

And giving free stuff creates a positive experience attributed to Kroger so you’ll be more likely to think positively about Kroger in the future.

Plus all that if they have your email\address they can send you more coupons or whatever to get you back in the door. And a more sophisticated system may track your spending to more effectively influence you through the first three mechanisms. But those are pretty classic ‘customer loyalty’ ideas from before big data was even a thing.

To get you to shop there more

I can kind of answer this question. For a year, I worked at a big box store where you can “save big money”. They had periodic training and one of the modules was on their store card.
The question was, “Why do we want people to have our card?”
I replied, “Because we don’t have to pay the standard processing fees.”
“No, it’s because of loyalty. If they have a [big box store] card, then they’re much more likely to spend their money here instead of [competitor].”

There’s a reason they are called, internally in business parlance, “Loyalty Programs.”

The point is to get you to come back. Using the Kroger card at Kroger gets you discounts. Or rather, gets you their regular price; their non-card prices are artificially inflated in most cases. So consumers form the idea in their heads that those discounts have value and will return to that store to take advantage of them. This is played up by the retailer in their marketing, who will use terminology to try to make you feel special about the program. (As if it’s not offered to absolutely everyone who will listen, and also everyone who won’t.) Look for words and terms like “exclusive,” “VIP,” or “members only,” or “just for you.”

The buy-stuff-to-get-rewards schemes are the worst, because they prey on the inbuilt sunk cost fallacy neurons in people’s brains. You are statistically likely to buy and spend more if you think you’re going to get something back, or spend a little more than you otherwise would have to meet whatever threshold they set to get the next reward. Even if the reward you get back does not actually match in value to the extra amount(s) you spent. (Hint: It never does. The house always wins.)

Nowadays, of course, they also track and record your purchase history and sell it to whoever will pay. Possibly anonymized in some way, but probably not.

The cost of the “free” stuff is covered by the profits they make on other products. It’s also offset by the data they’re selling about you.

The former is a classic prisoner’s dilemma, and with the way things are, taking part in the scheme is probably the better choice. If you don’t take part, you’re funding the people who do, at no benefit to you, unless enough people choose to opt out of the program to make it infeasible.

The latter poses a difficult problem. You can sort of work around this issue by taking multiple cards, providing a fake name for good measure (they’ll be more interested in your general area than your name in the first place), and most importantly, sharing and exchanging cards with friends and family to hide your personal data. Through some rather basic clustering analysis, these companies can still figure out your individual spending patterns if you’re sharing the card with a select few people (i.e. your household/roommates), but that’s an extra step they’d need to take that would probably only make sense if enough people would do it.

As (seemingly) nobody bothers with this privacy stuff, you’d probably be able to keep a bit of privacy by exchanging random cards with a small group of people.

I guess if they have a closer relationship to you as a customer you’re more likely to buy more of their products in the future.

Here, let Bill Burr tell you

They’re trying to capture the consumer surplus. Normally, a seller can have either high margins and low volume, or low margins and high volume. The retailers wet dream is to get the benefits of both. If the reward program profiles you as someone who buys coffee at $4.00, but not at $6.00, you’ll get coupons for coffee that the people who buy coffee every week regardless of price won’t get.

FWIW, I’ve found stores that don’t even have rewards cards frequently have lower prices than their competitors’ reward card sale prices.

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This is from a talk the CEO of the self-proclaimed first Canadian company to offer its own points based rewards card, not just AirMiles(which was presumably making money on the same thing, selling metrics.) it was a long time ago, so forgive the inexact quoting.

The purpose of the rewards card is metrics. With it, we know which customer is buying what, at what time, with what payment. This allows us to better target both sales as well as tailor entire stores to their respective customer bases.

Check out the Target story from years ago where a dad went mental that they sent vouchers for baby stuff to his pre 16 yo daughter.

Turns out they knew she was pregnant before the family.

Point being. Data is scary powerful at scale. Tesco in the UK is holding us hostage to get a card as literally half the items are cheaper with one. My friend group all use the same one though to pollute the data.