Historically, every time there’s been a need for two systems to communicate, a new protocol has been created (for example, SMTP for sending mail, POP3 for receiving mail, and the numerous protocols that database clients and servers use). The idea of web services is to remove the need to create new protocols by providing a standardized mechanism for remote procedure calls, based on XML and HTTP.
Web services make it easy to integrate heterogeneous systems. Say you’re writing a web interface to a library system that already exists. It has a complex system of database tables, and lots of business logic embedded in the program code that manipulates those tables. And it’s written in C++. You could reimplement the business logic in PHP, writing a lot of code to manipulate tables in the correct way, or you could write a little code in C++ to expose the library operations (e.g., check out a book to a user, see when this book is due back, see what the overdue fines are for this user) as a web service. Now your PHP code simply has to handle the web frontend; it can use the library service to do all the heavy lifting.