We have to carry 5 or 10% (Less in Europe) Contingency Reserve Fuel based on the required fuel from departure to destination, so for a long haul 747 flight that can be 10-12,000 kgs.
So if we find a destination along the route, lets say Gander for New York, we will dispatch to Gander with fuel that includes 5/10%.
We will have calculated the required fuel from that point to New York, if the weather conditions etc are suitable, the flight will reclear/re-dispatch from a point along the route to New York.
As the flight time is lets say 90 minutes, the fuel required is 12,000 kgs, 10% is 1,200 kgs which results in a fuel saving of 8,800/10,800 kgs.
You have achieved savings due to not carrying additional fuel, you might be able to carry additional payload, or you might have avoided a technical stop due to limited fuel.
Thats basically a KISS explanation..
Mutt