My guess is that it is for removing tiny clock gears/spacers/bearings. The various width slots accommodate different diameter shafts. As the plunger screws down on the end of the shaft the tool pulls up on the gear/bearing/etc gently slipping it off the shaft.

My SWAG is based on the fact that I made a tool for similar purposes some time ago.

Miniature hand puller

It's not exactly the same design but the function is similar.