The web browser has list of root certificates it trusts. These are public keys of CAs. The browser is saying you can trust that the private keys of these CAs are in fact private, and that anything that has been encrypted by one of those private keys -- including the alleged web server's certificate -- really came from the CA.
The certificate contains the public key of the web server and the web server's address (and company name etc.), encrypted by the private key of the CA. This encryption is done once, when the web site owner bought the certificate from the CA. After that, the web site owner keeps the certificate on hand to send you when you make an https request. Since your browser is able to use the CA's public key (which was already on your machine) to decrypt the certificate sent by the web server, and sees in the decrypted certificate that the certificate contains a host address that matches the https-serving host, the browser concludes that the host's public key (decrypted using the CA's public key) is authentic. The certificate routinely given out by the web host might still be coming from some random person spoofing the host, but at least you can be confident that it contains the authentic public key of the https-serving host you are aiming to communicate with.
You can then send data (like your credit card number) encrypted with the host's public key, and only the host's private key will be able to decrypt your data. No communication with the CA was necessary during the transaction.