Brushed DC motors have two wires because they require DC current (+VCC and Ground), while brushless motors require more complex driving circuitry which controls the AC current required to drive them.
Despite sometimes looking similar on the outside, the functional principles of brushed DC motors and brushless DC motors (aka. BLDC motors) are quite different.
These function by physical commutation, where the rotation of the armature breaks and establishes contact with pads which continue reversing the direction of current flow through the coil. This mechanical process perpetuates the cycle of magnetic attraction and repulsion which turns the motor.
BLDCs use three wires which are driven by the ESC with a phase-shifted AC waveform. Each wire's waveform is shifted by 120 degrees from the other two.
This is because BLDC motors are indeed three-phase AC motors, each with usually more than one actual coil per phase. Inside the motor, phases are wired up in ascending order, e.g. 1 2 3 1 2 3 ...