Why do rivers respond quickly to rain?

It depends on many factors. In urbanised areas, the surface run off is increased due to more impermeable materials (tarmac as opposed to a field). These impermeable surfaces don't allow as much (sometimes any) percolation, infiltration etc to happen. This means that the water travels faster by surface run off, therefore increasing the volume of water in the river. Areas of high density rain often make the river exceed the carrying capacity, and if sediment etc is washed in, the competence can be exceeded as well which can cause a flood. More water on a surface leads to more water into rivers. Take Boscastle in Southern England. I live near to Boscastle and in 2004 it flooded very badly due to the geology of the area (the rock type was highly impermeable), the urbanisation of the area, the fact that there is a confluence of 3 rivers there. There was also a large about of localised rainfall in the area in the weeks before saturating the ground. Its usually due to urbanisation and geology. There's many reasons, often a combination of them. I do A-level Geography in Dorset, further studying it at university. It's rarely just one thing.