"Trust this browser? If you choose to trust this browser, you will not be asked for a verification code the next time you sign in."

CharPatton wrote:

What is the extra security in sending the verification code directly to the browser? I could maybe see enhanced security if the code was sent to my iPhone, but sending the security code directly to the same browser seems kind of silly if I were nefariously trying to access my iCloud account from this browser.

The code is sent only to your own devices, so someone attempting to log in on another device won't get the code. It's assumed that you are protecting your devices with a passcode so that other people can't access them.

As Roger indicated, the verification code is only sent to your trusted devices using an encrypted communication channel (it uses the iMessage protocol.) It only appears to be sent to the browser. A stranger trying to log in to your account on an untrusted device will not be sent the verification code.

You can see a list of trusted devices associated with your Apple ID by going to you iCloud settings on any of your trusted devices.

As for why, after selecting "Trust" in the browser window, you continue to be asked to enter the verification code I'm not sure. Try resetting the browser by clearing the history. In Safari go to the Safari menu and select, Clear History…. Be aware that any time you reset the browser you reset the trust setting and have to select trust again.

I know it seems silly to send a verification code to the same device that you are logging in on but Apple sends the codes to all your trusted devices. That’s why it is recommended that you use passwords/passcodes to protect all your devices even if it is a desktop computer sitting on a desk at home. Also disable automatic log-in.

If you are consistently receiving that message try going into your iCloud settings and remove the MacBook from your trusted devices. When it gets added back in hopefully it will correct the problem.