If you've ever used an IBM Thinkpad pointing stick, you've used a strain gauge, the same technology that drives your electronic digital kitchen scale. 

(photo from wikipedia)

The pointing stick didn't move perceptibly. But it flexed a tiny amount. When something flexes, one side is compressed and the other side is stretched. So think of a diving board. When you stand on the unsupported end of the diving board, the board deflects, or sags. The top side you're standing on stretches around a  curve, the bottom side is compressed in a slightly smaller curve.  For the purposes of a scale, the diving board is called a beam. Beams in strain gauges are usually quite rigid in contrast to the diving board example. 

By knowing the physical properties of the beam, you could figure out weight by the deflection, much as spring scales work. But springs and materials change subtly  and become inaccurate over time. And you need quite a bit of motion to calculate things easily.

There are a few other physical changes that are more reliable we can take advantage of in this situation. And the changes can be very small, but still easily determined.  If you attached a copper wire to the diving board so it flexed and compressed at the same rate as the board, the wire's electrical resistance will change. The resistance changes because on the stretch side the wire gets longer and thinner which raises resistance. In compression, the wire gets shorter and thicker, lowering resistance.

In reality, a thin metal foil pattern is usually used. The metal is a special copper and nickel alloy whose electrical resistance is constant across a wide temperature range. Usually, electrical resistance increases with a metal's temperature. Being temperature insensitive is helpful in a kitchen setting. 

While this photo is not from inside a kitchen scale, it shows how such a foil is mated to the beam.

(photo from continuummechanics.org)

These changes in resistance are measured in an electrical setup  called a wheatstone bridge which can detect very small changes in resistance.  These changes in resistance are used to calculate the weight. The designer of the scale knows the physical properties of the deflecting beam and the corresponding resistance changes of the gauge. These two data points are correlated electronically in the scale to produce the weight readout. 

The strain gauge gives us thin, reliable and easy to use digital readout scales in the kitchen, the home and industry.