Think about it if the negative sign was not there. Then, the induced emf/mmf/flux/torque/speed, would be in the same direction as the existing quantity.
A motor is switched on, and the current is great. As the motor picks up speed, the induced current & voltage (mmf/emf) both increase. But the -ve sign, an artifact of Lenz' law, describes the induced quantities as oriented in a direction opposite to the original quantities. Hence the overall current drawn from the power source decreases since the induced current opposes the original.
Otherwise, it would increase, generating more flux, and yet more emf/mmf, yet more current, etc. Power would increase without bound and energy would be created out of thin air.
Lenz' law also covers mechanical quantities like torque and speed. If you pedal a bicycle type generator with no load, you reach a speed of let's say 300 rpm. Now add a significant load, 50 watts. The slight torque you applied pedaling under no load is now opposed by a counter torque due to the induction. When the load current is introduced, it generates a torque whose orientation is opposite to your pedaling torque. Otherwise the speed would keep increasing, with energy being created. In order to preserve the same 300 rpm speed (and the same voltage), you must increase your pedal torque so as to counter the counter torque. So an increase in electrical output power mandates a corresponding increase in mechanical input power.
Does this make sense?
Claude
