Why does the occupied channel width increase as you get further downstream?

This happens because in the lower course of the river, where the channel is much closer to base level, there is much more lateral erosion taking place and a lot less vertical. In the upper course, more vertical erosion takes place because the river is cutting down in an attempt to reach base level. As the river is reaching its mouth in the lower course, less vertical erosion is needed. Also because the river has a greater velocity downstream, due to more water entering the river from tributaries, there is more kinetic energy for lateral erosion to take place. This leads to more erosion in the form of abrasion, hydraulic action and cavitation which will widen the river channel.