How does Zillow get a database of all the sold homes in the United States?

Zillow pays 3rd party data collection companies to get the information from the public country records.  Some counties have their data in electronic form, that just needs conversion.  Other only have manual records in which case the data companies are manually typing the records they review.

Some of those data companies would allow you to pay them for access to the data too.  Some may insist on a regular periodic contract.  Some make the data they collect available on line to "subscribers".  Some may have a minimum requirement, or may not contract with non-business/government entities.  Obviously, getting enough provider company contracts together to cover most of the U.S. was one of the biggest challenges to setting up a company like Zillow.  It is also Zillow's largest expense.

If you just want select areas, you may be better off paying to be a subscriber to your local Multiple Listing Service.  It won't show all sales, but it would show any sales that were listed in the Multiples.  Zillow cannot sign up for those as typically only local licensed agents and brokers are accepted for membership access.