Why?
 

Why did Display Link suddenly pop up on my computer, evidently installing itself and rebooting my machine? I never bought or downloaded any software of this kind, so I'm worried this may be some kind of hijacking or identity theft. Please someone, explain what has happened and why!

DisplayLink software doesn't suddenly pop-up on its own and is unable to restart your computer without you saying so.

If you are part of a company, check with IT.
If you're not, then you will have connected a USB device using DisplayLink technology and Windows went to Windows Update to download the driver, which is normal process for any device.

DisplayLink driver is used in banking, by traders and an endless list of people who care about security.