Why does a long spanner make it easier to loosen a stuck nut than a short spanner?

This answer is based on simple mechanics. A spanner is a tool which you use to apply a twisting force on a nut. This twisting force is called a Moment.

The equation for a moment: moment = force x distance of application

As the force given by your arms will not increase just by using a bigger spanner, the size of the moment is directly related to the distance away from the nut the force is applied.

So with a short spanner, say 30cm (0.3 metres) long and a typical force from your arm of about 100N (about 10kg) the moment will be:

Moment = 100 x 0.3 = 30 Nm

Now if a bigger spanner is used, say 50 cm (0.5 metres) long the moment is:

Moment = 100 x 0.5 = 50 Nm

So you can see if you use a bigger spanner, you are putting a larger twisting force upon the nut, so it it easier to loosen.