When a bad guy steals your password, they could lock you out of your account, and then do some of the following:
Go through – or even delete – all of your emails, contacts, photos, etc.
Pretend to be you and send unwanted or harmful emails to your contacts
Use your account to reset the passwords for your other accounts (banking, shopping, etc.)
You'll enter your password
Whenever you sign in to Google or the SIL tile, you'll enter your password as usual.
You'll be asked for something else:
Insert and tap your Yubikey -or-
Use a code generated by Google Authenticator app on your phone (no network required!) -or-
Use your saved or printed backup codes.
Most people only have one layer – their password – to protect their account. With 2-Step Verification, if a bad guy hacks through your password layer, he'll still need your phone or Security Key to get into your account.