If you are a bird, sitting on a power line, you don't get a shock. (Lots of volts but no current can flow)
If you are on the ground (at ground potential) and you touch a live wire, with a potential very different from ground potential. (Lots of volts and lots of available current).
So what's this Earthing business? IF you connect all exposed metal parts of electrical equipment to Earth then they will all be at Earth potential. That saves you from ever getting a shock because nothing you touch can be at a high voltage. What's the catch? None. Only inside the equipment any fault which directly connects a live wire, internally, to the case will cause lots of current to flow and a fuse will blow. Other faults can cause a small current to flow from the live wire (Line) to Earth. These may not interfere with the operation of the equipment at all (there are many houses and electrical devices that are wrongly connected but no shocks have been delivered) but could cause a shock if the Earth connection were, somehow removed. It is possible to detect this Earth current, however, and to turn of the power (Power Breaker / Residual Current Breaker etc. etc are the names)
By the way, you don't need "electricity running through" something to get a shock. All you need, for a lethal shock, is high enough volts there and a few tens of milliAmps to be available to flow through you. There is a difference between Voltage and Current.