OK It turned out that I've must some program must have installed malware with it, which I uninstalled in programs and features and than I've reinstalled the browser and Speed Dials works correctly. Thank You all for your help.

It looks like this: http://imgur.com/a/K8axW

Today I was shocked to see something instead of regular speed dial. I am so shocked to see it, that I don't know whether I've installed wrong version of browser, or I have somethimg to do with settings, or whether it is a new "feature" and is going to stay forever?