Why does chilled equipment help when whipping cream?
Basically, cream whips better when it's cold. If cream were stored warm, the advice would be to cool the cream. Since it's already cold, the advice is to cool the equipment (as warm equipment will increase the temp of the cream). 

Mild science: the temperature effects the size of the bubbles that form, how fat clings to itself, whipping time, and overrun (air cell structure / air phase volume). In other words how light and fluffy it will get and how long it is likely to stay that way. 

Warning, heavy science from Influence of whipping temperature on the whipping properties and rheological characteristics of whipped cream:


  The effects of whipping temperature (5 to 15°C) on the whipping (whipping time and overrun) and rheological properties of whipped cream were studied. Fat globule aggregation (aggregation ratio of fat globules and serum viscosity) and air bubble factors (overrun, diameter, and surface area) were measured to investigate the mechanism of whipping. Whipping time, overrun, and bubble diameters decreased with increasing temperature, with the exception of bubble size at 15°C. The aggregation ratio of fat globules tended to increase with increasing temperature. Changes in hardness and bubble size during storage were relatively small at higher temperatures (12.5 and 15°C). Changes in overrun during storage were relatively small in the middle temperature range (7.5 to 12.5°C). From the results, the temperature range of 7.5 to 12.5°C is recommended for making whipped creams with a good texture, and a specific temperature should be decided when taking into account the preferred overrun. The correlation between the whipped cream strain hardness and serum viscosity was high (R2=0.906) and persisted throughout the temperature range tested (5 to 15°C). A similar result was obtained at a different whipping speed (140rpm). The multiple regression analysis in the range of 5 to 12.5°C indicated a high correlation (R2=0.946) in which a dependent variable was the storage modulus of whipped cream and independent variables were bubble surface area and serum viscosity. Therefore, fat aggregation and air bubble properties are important factors in the development of cream hardness. The results of this study suggest that whipping temperature influences fat globule aggregation and the properties of air bubbles in whipped cream, which alters its rheological properties.