What is the physical explanation as to why the kinematic boundary condition must hold at the free surface of a wave?

In reality, particles are perpetuately in motion; even due to Brownian thermic motion a particle will not stay on free surface. A free surface is an idealization; it is a collection of many particles that form a surface. However, if the particles on free surface are moving, clearly the free surface changes. Given an equation

$$f(x,y,z,t)=0$$

that characterizes a surface, then, after considering the same surface at a time $dt$ later, it holds:

$$f(x+v_xdt,y+v_ydt,z+v_zdt,t+dt) = 0. tag {*}$$

Particle motion must also be respected in equation (*). Using chain rule and Taylor expansion, you get:

$$f(x+v_xdt,y+v_ydt,z+v_zdt,t+dt) = f(x,y,z,t)+ sum_{i=1}^3 partial_i f(x,y,z,t) + partial_t f(x,y,z,t) = sum_{i=1}^3 partial_i f(x,y,z,t) + partial_t f(x,y,z,t) = 0$$