Here's a diagram:
The water enters at the top, percolates through the resin, and is forced up the tube which opens at the bottom of the resin bed.
As the water percolates through the resin on its way down, the ions (which are very mobile within the water) are drawn into contact with the resin where the ion exchange takes place.
As you can see from the diagram, the resin is actually a mixture of two different types to remove the +ve and -ve ions.