Why does my keyboard language switch on website password fields?
After doing a little bit of testing, I think it's because of this option is enabled in System Preferences > Keyboard > Input Sources:



It's checked by default and since cbs.com is in English, OS X will automatically change the input source to English (If you don't have English, or whatever language the page is in,  enabled, it seems it'll just select your default input source — i.e. the one at the top of the list in Input Sources).

I tested with Hebrew and Danish enabled and when focusing a textfield OS X changed my keyboard layout to Danish with auto-switch enabled and didn't with it disabled.

At least in my case, the swithcing wasn't limited to password fields specifically (which makes sense). The behaviour is the same in both Chrome and Safari.