How does Google know what you are going to ask when you start typing something in it anticipates what you are going to say with a selection of possible endings and what's this function called?

This is the auto-correct or auto-fill feature. Based upon all the searches performed in google, when you start typing a Word or phrase google tries to guess what you are searching for.

For instance, around the holidays the search "cookie recipes" becomes very popular. Therefore, when you start typing "cook", one of the suggestions will be "cookie recipes" because a lot of other people have searched for the same thing. However, when there was a big political story about Sesame Street earlier this year, "Cookie Monster" was a more popular search so it showed up in the suggestions. If there is a big story in the next few months about internet security, the phrase "delete cookies" may show up in the suggestions.