illegal attempt to access my web browser

I was sent a notification by Apple to activate two stage authentication on my account which I did. However I find this to be quite cumbersome.

I have been getting notifications on my iPhone and Mac Book Pro that someone in the South of England has been trying to access my web browser. Obviously I clicked 'DO NOT ALLOW.'

However it has taken a while to reset a password due to the two stage authentication process. Keep getting verification codes which has been cumbersome.

Can anyone explain whether this process is really worth activating? If it means receiving a v code every time it can be a pain and cost me for text messages every time.