The problem is they draw cool water upstream from the river and dump the warm water downstream. There's a limit to how much warm water you can dump because it raises the overall temperature of the water which impacts aquatic life.
Extreme heat and a severe drought do two bad things:
1. Lowers the river level. This threatens to expose the intake pipes drawing in cool water. Those pipes must remain submerged for their pumps to work. They don't rest on the riverbed because they don't want to draw silt from the bottom of the river.
2. Raises the temperature of the water. Lower water levels mean the river has less thermal mass and so the hotter air is warming up the water that much more. That means the cooling water is hotter. This typically isn't a problem as far as cooling the plant itself is concerned but consider that To - Ti (temperature of output - temperature of intake) is constant, so if Ti increases then To must also increase, which as already noted impacts aquatic life. The water may simply be too warm.
Either (1) or (2) is problematic - the two together are a big problem and threaten to shut down the plant. This doesn't just apply to nuclear power units either - coal units, IGCC units and HSRG units face the same issue as do nuclear units. You're right though, this is a big problem getting very little attention. This is one of many reasons why utilities are investing so much into solar and wind - no cooling required.