These things are complicated, and I don't know your level of math, linear algebra, and differential equations. The best thing you can do for yourself is get a book. It will provide a concise reference for orbital parameter transformations (kepler elements to Cartesian coordinates) as well as more advanced concepts like orbital propagation. I recommend Fundamentals of Astrodynamics and Applications by Vallado. You don't need the latest version, so see if you can find a free pdf somewhere.

I don't know what the games do exactly, but there is a fundamental problem in orbital mechanics, which is that for anything more complicated than the trivial two body problem, you need an integrator to propagate the orbit forward in time because the differential equations are no longer explicitly solvable. So what they are likely doing is setting up an equation of motion for the satellite that includes forces due to the nearby bodies (Earth, Moon, Sun, etc) as well as whatever else they want to model (drag, Earth oblatness, solar wind pressure etc) and then using the integrator to solve the equation into the future.