Imagine a large thermos flask with an immersion heater in it.
It has a vacuum so that's good as it does reduce heat loss.
I put in 1 kg of water in. It's cold - so I have to supply energy to boil it. Enough energy to change the temperature of 1 kg of water from 18c to 100c
It could cool down, although the vacuum does reduce the loss of energy.
But I want boiling water - so I keep the immersion on so it's constantly keeping the water at 100C - that's not a lot of energy .That's the standby mode of 10 watts - so about 0.24 KWH per day (6p at current rates)
You then take out 200g of water. You are left with 800g in. This water is replaced with cold water - so the temperature has reduced. You then have to supply energy to raise 1 kg of water to 100 C again. (This is the same amount of energy as you would need to raise the 200g of cold water to 100 C because....physics)
So you supply energy.
This is why it only costs 6p a day if it's just on standby mode and you don't take any water out. If you actually use it to get boiling water, you then have to supply energy to boil the water you replace it with.
Of course, you can just get the exact amount of water out - so that's good. It only works if you take out less than the minimum amount of your kettle. So 1 small cup. Otherwise, you would just be able to use a kettle.
If you didn't replace the water, then effectively you have boiled 1 kg of water, kept it warm using an immersion heater and then slowly used it all up. You then need to add more water and reheat it.
It's all to do with specific heat capacity, joules and kinetic energy.