Most of our rain comes from low pressure systems and we are familiar with those on our TV weather reports.This is a typical one approaching the UK.
The extra moisture in the atmosphere gets drawn towards the centre and concentrated where the rain falls and this gives an amount of water that would not otherwise be there.
There has been an increase in heavier rain events all round the world but not many go over the top and cause substantial floods and damage but as we head towards a 2C temperature rise and 14% more moisture in the atmosphere we should get used to the idea that one day a major flood will come to where you live.