The only purpose of a ground rod or a group of ground rods forming a ground field is to have a designed electrical path to dissipate a static discharge voltage (which can be lightning or other forms of static electricity) to the earth. When you ground electricity you will be using a ground rod in most cases.
If you place a current probe around the conductor going to earth or the ground rod itself you should never see any current flowing on the conductor. If current is flowing on the conductor going to earth a ground fault exists. Another term that could be used is leakage current. In either case there is a parallel path back to the voltage source through the earth that crates the loop for current to flow on the earth ground conductor.
Always remember current will always return to its voltage source.
AC will seek a path to the AC source, DC will seek a path to the DC source and a static discharge (lightning) will seek a path back to its source which is usually earth. If a return path is not designed, or multiple return paths exist, current will seek its own paths back to its voltage source.
When current seeks and finds its own path(s) back to its voltage source usually it will pass through equipment. It the voltage passing through that equipment is higher then its operating threshold that equipment will fail and an outage of some type will take place.