were all these, realized only by looking at Maxwell's equations? Or is there a physical reasoning behind them? For example, Why does the component of the electric field parallel to the surface of interface remain unaltered? I also read that the reason light bends when it passes through another medium is because only the normal component gets altered and the horizontal component remains the same(whereas the velocity gets altered because of the other electrons in the material that are driven by the source and produce a separate wave with a different phase and the superposition of these two waves seem to alter the speed of light in a medium $^\dagger$).
My question in short is, What would be my answer,if someone asked me, to explain the boundary conditions, without equations*.
*If that's purely based on equations, please ignore "without equations", but there's got to be something that's physically occurring which led us to create a model, right?.
$^\dagger$Is that right?