Fleming's left hand rule is used for electric motors, while Fleming's right hand is used for electric generators.
Different hands need to be used for motors and generators due to the difference between cause and effect.
In an electric motor, electric current and magnetic fields are present (which are the causes), and they lead to the force that makes the motion (which is the effect), and hence the left-hand rule is used.
In an electric generator, motion and magnetic fields are present (cause), and they lead to the creation of electric current (impact), and so the right-hand rule is used.
To illustrate why, consider that many types of electric motors can also be used as electric generators.
A vehicle powered by such a motor can be accelerated to a higher speed by connecting the motor to a fully charged battery.
If the motor is disconnected from a fully charged battery, and is connected instead of a fully flat battery, the vehicle will malfunction.
The motor will act as a generator and convert the kinetic energy of the vehicle into electrical energy, which is then stored in the battery.
Since neither the direction of motion nor the direction of the magnetic field (inside the motor / generator) has changed, the direction of electric current in the motor / generator has reversed.
This comes from the second law of thermodynamics (the generator current must resist the motor current, and influence the other to allow more powerful energy to flow from the more energetic source to the less energetic source).