Amazon market cap went up 3.5% as of the current tick, roughly $15.5B. The market effectively rewarded AMZN for the transaction far beyond the actual $13.7B tender offer!
This of course pales in comparison to the 478 million outstanding shares, so, you are correct: their shares don't appreciate nearly enough from the $15 billion surge in market cap to pay for the $13.7 billion purchase. However, to say "whatever the stock does has no impact on the company" is also inaccurate.
They do not, either in theory or in practice, own the assets of the corporation while it exists.
"effectively free" is a strange label to give this.
It means there's an opportunity cost: all other things being equal, it would cost Amazon two billion dollars in market capitalization to NOT do whatever the hell Wall Street THINKS they're going to do with Whole Foods.
Maybe the market thinks they're going to use them as bases for grocery distribution: reward, the cost of acquisition plus a bonus two billion. Maybe the market thinks they're going to gut the company, fire everybody, and then there's one less competitor for what Amazon means to do with groceries! Maybe the market thinks they'll take the existing Amazon grocery stuff and rebrand it Whole Foods and the rabble will buy into that for a time.
The point is that this money really isn't free: it's conditional on whether Amazon does what the investors think they're gonna do, with the acquisition. If Amazon fails to do that, the market capitalization can evaporate.
My own hunch is that the market thinks Amazon is super efficient and acquisitive, and that Whole Foods is weak and inefficient and misguided. As such, the expectation of greedy and amoral investors is that Amazon will take over and whip Whole Foods into shape, maybe even fire everybody and replace 'em with Amazonians, and that will make Whole Foods 'better' and justify the exercise. They certainly aren't buying into Whole Foods' 'enlightened' memes: that has no place on Wall Street.
So I'm guessing it's BECAUSE Whole Foods has been 'enlightened' that the prospect of wrecking that so motivates investors. They figure that the Amazon way is clearly the winning way, and they figure there will be a BIG difference in how things are done, and it's a difference they characterize as 'good for capital', so there's a synergistic effect that makes it 'free' for Amazon plus a bonus. And then Amazon is obligated to do what's expected of them, which I think is probably kinda predictable.