Recently I used transit-finder.com to get a really cool picture of the ISS transiting the sun. The prediction from the website was spot-on. Since then I'm absolutely fascinated by how the prediction works. I have a background in engineering and math but not really sure where to start in order to make my own flyover or transit predictions. I see that there's plenty of freeware and websites that make these predictions but I can't find anything that really points me to the math behind what's happening. Does anyone know where I can find some sample problems/calculations or maybe something that walks through this process?
I guess more formally my question is: given the TLE data from a given satellite, how can I predict future flyovers of my location on my own?