A drainage basin is the area drained by a river. It is the area that the river gets its water from, andthe river reaches out to all parts of its basin using tributaries (also called branches) that resemble the veins on a leaf.
All of these tributaries feed water into the main channel, so the main channel gets a very large amount of water.
But most water does not reach the tributaries directly as rain for, as you can see, the river channel is just a tiny part of the drainage basin. Most rain falls on grass or forest and then seeps into the soil. This water than makes its way down valley sides and seeps into the river through its bed and banks - quite unseen by us. This is the water that helps a river keep flowing in times when there is no rain.
Just occasionally, the soil fills up with water (for example after weeks of heavy rain in winter) or the rain cannot seep into the soil fast enough (as with a summer thunderstorm). Then the water flows over the surface. It gets to the channels far faster when it flows over the surface, and this is when floods are likely.