What does "Trust this browser and sign out" mean

It means that, if you trust the browser, you shouldn't be asked for a verification code the next time you sign in. You still have the option of just signing out if you prefer.

cqskie wrote:

It asks this question every time I sign out of the same browser from the same login id. If it is a trusted browser, why does it ask me that every time?

Isn’t trusted trusted?

Technology ain't perfect. If you remove or don't accept cookies, that would be one reason it doesn't "stick."