Okay, turns out Google is my friend and although I'm not smart enough to figure it out on my own, I did manage to read the product page from the company and it clarified the use for me.

The squeeze bottle that comes with the set has a removable syringe.  If you remove the syringe by twisting the little plastic base, you can the put the glass end of the TNF into the squeeze bottle to use it to draw solvent upward into the tube.

This is the page that has the information.  http://www.flex-i-file.com/touch-n-flow-system.php

For what it's worth, I think this is a really cool little solvent applicator.  I wish there was one for CA glue, too.  I am using a tapestry needle with the eye filled partially off (leaving a Y shape) to apply CA right now and it works pretty good.  But, this TNF is very convenient and makes using plastic solvent so much easier.