The terms that sound like you’re already been screened and approved should not be taken literally. They're marketing language. They come with a standard application and do not constitute any type of actual pre-approval.
Credit-card offers that say you're pre-approved, pre-screened, pre-qualified, pre-selected – or any other “pre,” for that matter – are known as "invitation to apply" (ITA) offers. They mean that whoever made the offer – a bank, retail store, or other lender – has done a background check that might include your credit score, borrowing history, and any other personal information available to them. The bank or credit card company may have purchased a list from one of the credit bureaus that fits the criteria for people they are targeting. With that information, they decide if you meet their criteria for credit.
Because they already evaluated you, chances are you will qualify for that offer. But credit laws require the card issuer to assess your ability to pay before approving you for a card. And you have just as much chance of qualifying for hundreds of other credit cards you can find online through credit-card comparison websites like mine. The best online offers frequently represent a better deal than what you get in the mail.
So if you are interested in the card, look beyond the marketing terms and assess the terms of the credit-card offer.