Well, to collect data from a mainframe. That's one reason why some people use screen scraping. Mainframes are still in use in the financial world and often it's running software that has been written in the previous century. The people who wrote it might already be retired and since this software is very critical for these organizations, they really hate it when some new code needs to be added. So, screenscraping offers an easy interface to communicate with the mainframe to collect information from the mainframe and then send it onwards to any process that needs this information.
Rewrite the mainframe application, you say? Well, software on mainframes can be very old. I've seen software on mainframes that was over 30 years old, written in COBOL. Often, those applications work just fine and companies don't want to risk rewriting parts because it might break some code that had been working for over 30 years! Don't fix things if they're not broken, please. Of course, additional code could be written but it takes a long time for mainframe code to be used in a production environment. And experienced mainframe developers are hard to find.
I myself had to use screen scraping too in a software project. This was a scheduling application which had to capture the output to the console of every child process it started. It's the simplest form of screen scraping, actually, and many people don't even realize that if you redirect the output of one application to the input of another, that it's still a kind of screen scraping. :)
Basically, screen scraping allows you to connect one (web) application with another one. It's often a quick solution, used when other solutions would cost too much time. Everyone hates it, but the amount of time it saves still makes it very efficient.