What is the purpose of measuring prospective short circuit current and earth electrode resistance?

Sooner or later an electrical appliance will experience a short circuit.  Either due to a lifetime of stresses or simply due to a faulty component.  It is necessary to determine that:

A.  As a result of the worst case short circuit condition that a product will not become a fire or fragmentation hazard.  This is usually covered by evaluation to an appropriate regulatory safety standard.

B.  As a result of a worst case short condition that may create current slightly below the operational value of a circuit breaker or fuse in the power distribution path, that said path is capable of withstanding that current without causing damage to the wiring which may result in a fire hazard.  To allow for the different properties of different types of a safety disconnect, standards may add margin by having the path evaluated to 125%-200% of the breaker rating.

For the Earth electrode resistance, it is important to determine under a potential fault condition that the earth connection is of suitable ampacity to safely carry potential fault currents without opening.  Also, be of low enough resistance that while carrying those fault current it will not result in voltages that may be hazardous if touched (as many metallic enclosures on appliances may be grounded.)

You don't measure prospective short circuit current, you calculate it. Its the most current that could flow in the event of an earth fault. It is based on many factors depending on the distribution network. You need to know it so that you can co-ordinate all of the protective devices and not have equipment potentially subjected to currents that it may not withstand

Earth electrode resistance is measured and is the resistance an earth electrode has with the earth. This measurement usually has a maximum value depending on the regulations you have to adhere to. The regulations will state is maximum. To measure it, you place two test rods, one at twice the distance of the other from the electrode, then measure and calculate the resistance.