Why does the main circuit breaker trip but not the smaller branch circuit breakers?

If the main breaker has ground fault detection, and the fault is a ground fault, then only the main may trip.

Also if the fault is a direct short, the fault current may be several thousand amps, much greater than the trip point of both breakers. Breakers have an I^2T curve (current squared-time), which is an indication of how fast the breaker will trip at a given overload. When both breakers are overloaded, the breaker with the faster I^2T rating at that current level will trip first. This would be an indication that the breakers are improperly specified or adjusted.