When people talk about a device "drawing" current, what do they mean? Why do devices under load "draw" more current?
Think of it as "drawing" extra breath whilst jogging as opposed to walking.

A circuit under normal conditions will appear as a certain impedance. For instance, a DC motor operating without a mechanical load will spin at a rate determined by the number of its windings, contacts, permanent magnets etc. As a load is applied to the shaft, the rotor decelerates, reducing the impedance of the windings being contacted. Simplistically, the impedance is determined by the speed (frequency) at which it spins. As the windings are inductive, the reduction in angular frequency reduces the impedance. As a result, the current increases, so it "draws more breath", so to speak.