Kinetically constrained models (KCMs) are models of glass formers based on the concept of dynamic facilitation. This concept accounts for many of the characteristics of the glass transition. KCMs usually display a combination of simple thermodynamics and complex glassy dynamics, the latter being a consequence of kinetic constraints. Here we show that KCMs can be regarded as systems whose configuration space is endowed with a simple energy surface but a complicated geometry. This geometry is determined solely by the kinetic constraints and determines the dynamics of the system. It does not affect the overall distribution of states. Low-temperature dynamical slow-down is then a consequence of the competition between the paths allowed by the geometry of configuration space and those leading to energy relaxation. This competition gives rise to dynamical bottlenecks unrelated to static properties. This view of the dynamics is distinct from that based on an underlying static rugged energy landscape. We illustrate our ideas with simple examples.