(How) does blocking a (servo/DC) motor break it?

Simply stopping a motor from turning does not by itself hurt it. Think about it. The motor is stopped when you first apply power, and nothing gets hurt.

Most motors are designed so that the mechanical forces from the maximum torque won't hurt the motor.

The reason some motors shouldn't be stalled with full voltage applied is heat. All the electrical power going into the motor goes to heating the motor. Many motors can be over-heated this way. Designing them to be able to dissipate the heat from maximum applied voltage while stalled would require making other parameters less desirable. In many cases that's not worth it.

What makes stalling a motor even worse is that's when it draws the most current. So not only is more electrical power being dumped into the motor, but a higher fraction of that is turned into heat in the motor at low speed.

To a pretty good first approximation, think of a motor as a resistor in series with a voltage source. The resistor is just the DC resistance of the windings. The voltage source represents the motor acting as a generator as it turns. The voltage is proportional to speed, and opposes the applied voltage when the motor is turning due to this applied voltage. A stalled motor therefore only looks like a resistor. As the motor speeds up, that resistor is still there, but less voltage is effectively applied to it, thereby causing it to draw less current.

Sophisticated motor controllers either model the motor's internal temperature, or measure it outright. This includes tracking how much of the power delivered to the motor is transferred away by the rotating shaft, and how much is dissipated by the motor as heat. When the motor gets too hot, the drive power is reduced to avoid damage.

Since the damage is due to heat, stalling a motor under full power is OK for at least "short" intervals. How short "short" is depends on the design of the motor, and is something that good datasheets give you guidance on.