They get three things out of it; data, loyalty and influence.
When a company can identify your shopping behavior across many visits they can build a much more accurate profile of you. When they can tie that to purchases with partner vendors they get an even better profile. If they can tie that purchase information to real world contact information it gets even better. It's like being able to follow your customer around all day to get to know them better.
The loyalty programs also provide a way to invoke the sunk-costs fallacy. Once people get a hole punched in your coffee card they feel like they've earned part of a coffee that they'll miss out on if they don't get the rest of the holes punched. So they keep coming back (and they'll often get an other hole punched with the free coffee).
Companies can also use the loyalty programs to promote new products, steer customers to more profitable products, disguise shrinkflation and target their ads more effectively.