You just transfer the money from your bank account to the bank account of the merchant. There are various methods to do that, I prefer giropay[1]. If it's a merchant you regularly use like amazon you allow the merchant to directly withdraw[2] the money from your account.
It's not like this is some weird highly advanced stuff. Surely that's possible in the US as well?
It is not generally possible to do this in the US. Amazon allows it, but most places do not because it's not a very in-demand service — using cards is safer and easier than giving out your bank information, so not very many people insist on the latter.
How could a bank transfer be less safe? I don't know the US system but if I give out my bank account number the only thing someone can use it for is to send money to me.
On the other hand if someone malicious gets my credit card number they can use it to buy stuff and I have to go through a lot of trouble to get a new card.
It's somewhat the opposite in the US. With your bank routing and account number someone can make out bad checks or EFT pull from your account. Halting or reversing this can be quite difficult and in some cases it isn't possible to get your money back.
With credit cards the liability is on the merchant that accepts the card to verify that the person presenting the card is authorized to use it. Card issuers will reverse charges and leave the merchant out the money in the event of fraud.
It's not like this is some weird highly advanced stuff. Surely that's possible in the US as well?
[1]: http://en.wikipedia.org/wiki/Giropay [2]: http://en.wikipedia.org/wiki/Elektronisches_Lastschriftverfa...