These particular instructions were written for the McGill word tag punch, a large punch with multiple cuts being made at once, but most of the concepts will remain the same for other punches.
A really good tip for brand new punches is that you need to punch through wax paper several times before trying it on paper. Do this every so often as maintenance also (adding a routine of punching through tin foil now and then to sharpen) to keep your punch gliding through the paper better.
Back to the problem at hand. Pull out all the paper you can, both from the edges and the punched pieces themselves (use a needle tool to try to grab at these). Place the punch upside down while you're doing this so you can see where you are working. Now. It's "catching" somewhere - press down as hard as you can (from the back). I try to kind of "bounce" on it to see if it jars it loose. Next, try a screwdriver. You want a nice strong screwdriver. Use it to push the grooves down as you pull the base up (a second pair of hands really comes in handy here if you can find some!). As a very last resort (seriously, this is a PITA - don't try this unless you have some time, patience, and a good memory), there are two screws holding the base unit together. You can unscrew these (and if you're lucky a little loosening will pop the punch back into shape without having to take it apart) and remove the back to unstick everything. Just be careful to remember exactly where and in what order everything went so you can put it back together! Once you get it unstuck, definitely do the wax paper and tin foil (I would recommend wax, then tin, then wax again) and then try again. Another tip to using this punch (and other multiple cut punches this size and larger) is to make sure you are punching on a flat surface and using both hands to press down evenly on all sides. The springs (at least on mine) were really stiff in the beginning too. Once I used it a lot, it loosened up a bit.
Punches are a great tool but they do require some routine maintenance! I keep wax paper and foil with my punches and on punches I know give me issues, I do this every time I use them. Others just do every so often.