Once you have entered the information into an online payment form, it is carefully encrypted and sent to the appropriate server for processing - again, no human is necessarily viewing it.
When using a proxy website that acts as sort of a "browser within a browser," you are using an encrypted connection to the web, which "anonymizes" the computer user and prevents the IP address from being recorded.
However, many people are still afraid to buy swimwear online, so it is handy to mention that most reputable vendors provide secure encrypted sites that minimize the chances of your personal and/or credit card information being released.
This is a very safe thing to do, but it's also important to make sure that the connection to the bank is through a secure and encrypted protocol.
Most companies provide a secure, encrypted environment for your credit card number and have layers of security to prevent unauthorized attacks.