The jet stream
Jet streams are high altitude rapidly flowing currents of air. The energy to drive the is obtained from the temperature difference between the cold polar and warmer adjacent regions. The jet stream is a very significant driver of weather and the role of the polar jet stream in the Arctic is increasingly understood in driving recent extreme weather.
As the temperature gradient driving the jet stream weakens the behaviour of the jet stream changes. The track it describes across the planet becomes more erratic with higher amplitude of movement. Since the jet stream acts to segregate colder Arctic air from the warmer adjacent air (to the south in the Arctic) this enables the movement of air that is hotter or colder than usual to regions that would previous not have experienced it or that would experience it less often.
Another aspect of the changing behaviour of the jet stream is that the progression of the weather systems from east to west becomes slower and it becomes possible for the jet stream to become "stuck" in a given region for prolonged periods of time. This means that the weather it is bringing at that point in time will effectively be magnified. If it is bringing heat there is a much greater chance of a heatwave and if it is bringing rain a much greater chance of flooding.
The reason that we see a much greater chance of an extreme event is that most weather becomes extreme if it persists. For example if we have more heat than usual this increases the rate of evaporation of moisture from the soil or transpiration from trees. These processes act to cool down the ground by absorbing energy to evaporate the moisture. However as the amount of moisture available for evaporation decreases more of the energy being delivered by the sun goes into increasing the temperature - resulting in a heat wave.
Where rain becomes a persistent pattern it also tends to increase the probability of flooding as it persists. This is because the ground becomes saturated and further rainfall tends to run off instead of being absorbed into the ground.
Therefore what at first may appear to be a rather subtle effect of climate change and sea ice loss is actually fundamentally important to agriculture as we rely upon thes predictable (within certain bounds) weather patterns to produce the food that our civilisation relies upon.
We are already experiencing significant changes in the jet stream at the current state of ice loss (2012) and the effects of losing the ice entirely (even if only during the summer at first) could reasonably be expected to be dramatic far beyond the extreme weather we have experienced recently.