Here's an intuitive approach.

Consider the source of the vibrations in the middle of the medium. It's a combination of the inertia of the particle and the forces applied by neighboring particles.

When a particle at the boundary vibrates, the properties of the particle at the different medium applies a different force to the boundary particle. This changes the oscillation period of the boundary particle, which in turn applies a different force to the other particles in the original medium.

Essentially, the boundary particle goes out of sync, which reads as a reflection when scaled up to the entire boundary