How does boundary condition and phase change on refraction related?

Light is composed of electric and magnetic fields that are perpendicular to each other. The tangential component of electric and magnetic fields are continuous across boundaries. If we use these continuity relations as well as the equations relating the speed of light $c$ and the index of refraction $n$: $$frac{E}{B} = c = frac{c_0}{n}$$ a very useful set of equations known as Fresnel's equations can be derived. A phase shift of $pi$ is represented by a flipping of sign of the electric field across the boundary.

I often answer this question with the analogous situation of a travelling wave on a string. If the end of the string is tied down, the reflected wave flips phase. This is analogous to an interface where the 'outer' material has greater index of refraction.
If the end of the string is free to move, the reflected wave holds phase. This is analogous to an interface where the 'outer' material has a lower index of refraction.
See this exact discussion here