These show up periodically. I can't remember when it was patented but it was in the late 19th-century, as far as I remember.
It's a form of traveling pen. There were hundreds of different types of pens which could retract the nib back into the pen for carrying around in your pocket. Some of my favorites were often called Water Pens. They used special nibs which had a hard pill of dried ink usually carried underneath the nib. All you needed was a glass of water and you had pen and ink at your fingertips. These had a certain run of popularity.
Eagle only started making steel pens themselves in 1891, but they had been making penholders (as well as the pencils which gave the company it's name) for a while before that.
For this penholder, you'd need a fairly small and narrow pen, like a Gillott 303 or 404. You probably couldn't use it with something like a Falcon unless the tube is wider than it looks.