That little rounded triangle thing on the arm of the pin is a "bubble guard". You push the pin's post through your shirt, slide the bubble guard on ("behind the shirt") and then continue pushing the pin out the other side of your shirt. Fasten the clasp as normal.
I'm with icelandelf, it sounds like you may actually be having a problem with the clasp itself. If that's the case, then you would need to take it to a jeweler for repair or replacement.
Of course, the whole point of a bubble guard is to keep the badge from falling off if the clasp fails.
The guard "grips" the pin post and acts a little bit like an earring backing. If the pin clasp fails, the pin post can't slide off the shirt because the bubble guard is on the other side and, well, the guard is too big to pass through the shirt.
ETA: This is what the back of your pin looks like. You said you simply push the pin down to close it. I wonder if this style has an "auto-clasp" system where the inner piece folds in over the post when the wearer pushes down on it. Any GC Kappa Sigs want to clarify?
.....Kelly