A DC motor is called a “self starting machine” because when it is directly connected to a rated voltage DC power supply, it does not require any external assistance or additional starting equipment, and its rotor can automatically start rotating from a stationary state.

1. Core principle: Law of Electromagnetic Force
The basic working principle of a DC motor is that the current carrying conductor is subjected to electromagnetic force (i.e. Ampere force) in a magnetic field.
The direction of force is determined by the direction of the magnetic field and the direction of the current in the conductor (following the left-hand rule).

2.Why does “self starting” require special design?
If it is just a simple single coil in a magnetic field, it will encounter a problem: dead center.
When the plane of the coil is perpendicular to the direction of the magnetic field (i.e. neutral plane), although there is no current in the coil (or the brush is short circuited between two commutator segments), even if there is current, the forces acting on both sides are equal in magnitude and opposite in direction, and act on a straight line, forming a torque of zero.
At this time, the motor cannot start and needs to be manually turned to continue rotating, which is called non self starting.

3. The key to achieving “self starting” of DC motors: brush and commutator
There are many coils (windings) on the rotor (armature) of a DC motor, and a commutator is installed, which, together with fixed brushes, solves the dead point problem.
Structure: The commutator is a cylinder composed of multiple insulated arc-shaped copper plates, each connected to the endpoint of a coil.
Electric brushes (usually graphite blocks) are fixed and pressed onto the commutator.
Function: When the rotor rotates, the coil rotates and the commutator also rotates.
But the electric brush is fixed and immovable.
This achieves the function of automatically switching the direction of current.

4. Decomposition of self starting process
Assuming a simplest two pole motor (a pair of magnetic poles, a rotor coil):
Initial position (any position):
When the direct current is connected, the current flows into the rotor coil through the brush and commutator.
If the coil is exactly at the dead center (geometric neutral line position), that is, the coil edge is exactly in the middle of the magnetic pole.
At this point, although the torque of a single coil is zero, modern motors overcome this problem by placing the electric brush on the geometric neutral line and making the pole arc width of the magnetic pole slightly larger than the span of one coil, or by having multiple coils on the rotor.
In this way, even if the coil is at a dead point, there will always be one or more coils deviating from the dead point due to the edge effect of the magnetic field or the presence of other coils, which can generate torque.
Generate initial torque:

Assuming the coil is not at the dead point.
According to the left-hand rule, one side of the coil is below the N pole and the other side is below the S pole.
Due to the fact that the direction of current is determined by the electric brush and commutator (the current introduced through the electric brush always flows in from the wire below the N pole and flows out from the wire below the S pole), the electromagnetic forces acting on both sides are in the same direction, forming an electromagnetic torque that drives the rotor to start rotating.
Maintaining unidirectional torque (thanks to the commutator):
After the rotor rotates 90 degrees (half a turn), the coil edge that was originally under the N pole will turn under the S pole.
At this point, if there is no commutator and the current direction remains unchanged, the force acting on that side will become in the opposite direction, causing the torque direction to reverse and the motor to stop (which is one of the reasons why AC motors cannot start automatically when single-phase).
However, DC motors have commutators!

When the coil turns around the neutral plane, the commutator connected to the coil also turns around, and the commutator that was originally in contact with the brush on this side is replaced with a brush of another polarity.
This means that regardless of which side the coil rotates to, as long as it is at the N pole, the direction of the current flowing is fixed (such as always flowing inward), and at the S pole, the direction of the current flowing is also fixed (always flowing outward).
The result is that the polarity of the magnetic pole at the edge of the coil changes, but the direction of the current also changes synchronously. According to the left-hand rule, the direction of force on both sides remains consistent, and the direction of torque remains unchanged.