Well they are obviously using this API:

Google Maps API - Autocomplete for Addresses and Search Terms

You can use autocomplete to give your applications the type-ahead-search behavior of the Google Maps search field. When a user starts typing an address, autocomplete will fill in the rest.

In order to use autocomplete, you will need to load the Google Places library using the libraries parameter in the bootstrap URL for the Google Maps JavaScript API:

<script type="text/javascript" src="https://maps.googleapis.com/maps/api/js?libraries=places"></script>

How to return cities only

Note that by default it is an address lookup, not a city lookup. To restrict it to return cities only, use an options argument for the Autocomplete object constructor. The (cities) type collection instructs the Places service to return results that match either locality or administrative_area3 (this means "cities" of certain size, i.e. not a remote rural cottage village)

var input = document.getElementById('cityLookupTextField');
var options = {
  types: ['(cities)']
};

autocomplete = new google.maps.places.Autocomplete(input, options);