I am having trouble understanding how this circuit works. I've breadboarded it out and the LED flashes but I can't seem to wrap my head around how the NPN vs the PNP is creating the flash. Any help is greatly appreciated!

 

And I'm having trouble understanding your drawing.

Standard schematics are the language of electronics.
Please draw it with standard schematic symbols (below), and with the power connection at the top and ground at the bottom.

 

It is a little complicated in part because parts are used in strange ways. In my drawing I used three diode in place of a LED because I could not find one that is right. I made C1 small so it will flash faster. Other wise it is about the same circuit.
Once it is up and running....
Q2 base is slightly below ground. R1 pulls up on the base but must charge (or is it discharge) C1. So Q2 base voltage slowly moves up to around 0.6V where Q2 turns on. Q2 then turns on Q1 which pulls up on the LED very hard. Q1 soon can not pull that hard and opens up (gives up). Now Q1-C voltage drips which (through C1) pushes Q2-Base down below ground. and turns off Q2 and thus Q1.