"Symplectic" means that the physics calculations are approximately energy-conserving, in that the system doesn't gain or lose energy over time. This is incredibly important for orbital calculations, because orbital energy relates to the shape of the orbit.
If you load up stock KSP right now and visit a ship in orbit, you'll notice with KER or MechJeb that the orbit's periapsis and apoapsis very slowly decay with time. That happens because the stock physics integrator is not symplectic. You don't see a problem with time-warp because stock KSP runs "on rails" based on Kepler's laws, without actually simulating the N-body forces.
Surprisingly, "symplectic" has little to do with the overall accuracy of the calculations. You can have very accurate calculations that exhibit energy drift, and you can have very inaccurate calculations that conserve energy.
"Partitioned" refers to a detail of the simulation, in how it separately updates position and velocity. You don't have to care about that one, it's an internal thing.