That would be something from your yahoo email settings causing that verification or your using a unsecured ISP provider for which yahoo detects and wants and verification of the user is whom they are. And if this happens at work that would be a IT related issue but if from home then you got some ISP problems or addon issues that is being detected as problem resulting yahoo email asking for verification. Otherwise if this continues then you need to have a computer shop look at what is happening to your email login problems.

A detail like a website remembering you (log you in automatically) is stored in a cookie.

You can use these steps to make a website recognize and remember you:

• you can create an Allow exception with the proper protocol to keep specific cookies, especially in case of secure websites and when cookies expire when Firefox is closed.
• Options/Preferences -> Privacy & Security
  Cookies and Site Data: Exceptions