In the United States, Redfin is organized/licensed as a real estate brokerage and hence has access to MLS (multiple listing service) data. Movoto's home page says they're a full service brokerage - I haven't used their site and don't know much more about them, but I imagine they've got the same MLS data.
Zillow is not a brokerage and does not have access to MLS data, they've developed their own database from other sources.
Typically, to get access to the MLS data for a given state or region, you need to be licensed as a real estate broker/agent or as an appraiser for that state/region.
In the real estate industry, the term for a website that includes MLS data is "virtual office website" or VOW. There has been quite a bit of fighting and arguing and drama within the real estate industry about sharing data and setups which allow Broker A to build a website which shows Broker B's listings, even though in bricks-and-mortar world, Broker B shows Broker A's houses all of the time, and that's what MLS was set up to facilitate.
Another source of data is public records - typically, when property changes hands, a deed or other document is recorded in the county recorder's office. The deed is public record, and public record companies copy, index, and resell that data. In California, it's usually possible to back-calculate the sale price by looking at the documentary transfer tax paid to record the deed, so transaction prices are for the most part public records, too, as are mortgage amounts. This is where Zillow gets their data about historical transactions and trends.