A “Driver class” is often just the class that contains a main. In a real project, you may often have numerous “Driver classes” for testing and whatnot, or you can build a main into any of your objects and select the runnable class through your IDE, or by simply specifying “java classname.”
Without context, it’s hard to tell. Is it talking about a JDBC driver, perhaps? If so, the driver class is responsible for implementing the java.sql.Driver interface for a particular database, so that clients can write code in a db-agnostic way. The JDBC infrastructure works out which driver to use based on the connection string.
If the book wasn’t talking about JDBC though, we’ll need more context.