The law must be interpreted as narrowly as possible. The only thing this ruling does is prevent a store from barring the use of an external payment processor. That’s it. It does not prevent a store from taking a cut according to its contract from any transactions from a game hosted on their store. In practice this means no change since most developers won’t want to pay a cut to the store and then a cut to the payment processor. Furthermore, this ruling is based off of California law, so a store can reject all developers from California if they don’t want to deal with it.

Ok, I hope you are right and that's how it is going to be.

Meaning, nothing really changes? Epic wanted to keep their free-to-play game on the Apple platform, without giving Apple any cut from the income Epic gets from that game... but now they can't do that anyway, without giving a cut to Apple from all the microtransactions?

I wonder though, how can Apple keep track of all the payments made using external payment processors, so that they get the cut from all purchases that they are eligible for?

As tremere110 said, the law doesn't force game stores to host any game they don't want for any reason, it just prevents them from blocking external payment processors for games that want to use them for IAP's. It won't affect GOG at all since GOG (thankfully) doesn't host any of those. Don't worry, GOG won't be forced to host Fortnite (also thankfully)... ;-)

There is a reason Epic was ordered to pay Apple. They violated their contract to give Apple 30% of their sales from the game they sold on the App Store and were ordered to pay the amount owed. The judge viewed the contract as completely valid.

You can put whatever you want in a contract and as long as both parties are capable of consent, most judges are hesitant to interfere with its terms without other serious factors - like one party having a monopoly and enforcing total control over the market (which the judge ruled against in Epic's case). Don't like the terms of the contract then don't agree to it and therefore don't do business with each other. California anti-steering law prohibits Apple from banning a developer from sending their users to another website - so that can no longer be a requirement going forward for Apple.

How can Apple track these transactions? However a new contract from Apple stipulates essentially. They could require the other party turn over any transaction records regarding a particular App and the third party payment processor. Sure the other party could lie about it - but it will be handled as per whatever contract is drawn up.

In many cases they will get caught in the lie eventually. In the US for example if there is a discrepancy between what Apple earned from Epic and from what Epic earned the IRS will notice and get involved. Epic isn't going to want that to happen. Similar Tax enforcement agencies in other countries would be scary even if they only had half the power of the IRS. So no, Epic will not be lying to Apple about how much they owe.

Epic didn't win anything here except a slight technicality that really doesn't benefit them in anyway. Apple will still fight against it since they're a walled garden whose users tend to prioritize safety over freedom - and getting redirected to websites Apple has no control over is a security issue for them.