In simple terms, the strain gage is one of four resistors in a wheatstone bridge circuit. The strain gage is the variable "resistor" with its resistance being a function of the strain that the gage undergoes.
A simple and common type of gage is the bonded strain gage. It is glued to the surface upon which strain is to be measured. When the surface undergoes strain, the strain is transferred to the gage which undergoes a resistance change that is measured by the electronic circuitry that it is wired to.
Current technology simplifies the setup and makes the connections fairly easy...just a couple of solder connections.
Have done this many, many times on dynamic structures (amusement rides, industrial equipment, loading equipment, treestands, etc.)