In the case of the BannerCoin ICO -
bannercoin.com which is accepting currencies other than Ethereum for the token sale, users register and include their Ethereum address for where they want the tokens to be sent. Once they checkout and go to pay for the tokens, the system will generate a transaction for that user. Once payment is received successfully the record for that registered user is updated to show that their payment has been processed successfully. Coins will then be sent to their registered Ethereum address.
This process is different than relying upon a smart contract like many ICOs since various cryptocurrencies are being accepted rather than only Ethereum.
This same process will be how checking out on an ecommerce enabled website will work. The difference being that tokens are being sent to a registered address as opposed to physical goods. BannerCoin is a payments platform.