When you'll encounter a CAPTCHA
Creating a new Yahoo account.
Performing too many failed sign in attempts.
Forwarding messages (chain letters, jokes, etc.).
Accessing your account from a new browser, computer, or location.
Sending an email containing HTML, links, embedded graphics, attachments.
The purpose of it is simple.
Yahoo uses CAPTCHA verification to protect against spam or other unauthorized account access. By selecting the correct images, you verify you're human and not a spam-sending computer.
Do I mind it? No, not really, I seldom see it and it has stopped a lot of spam.