Chilling the batter slows various processes and conserves certain conditions so as to promote the production of many small, thin-walled, crisp bubbles in the cooked batter. For example chilling reduces the rate at which dried protein and starch particles (mostly amyloplasts) and plant cell walls in the mix imbibe water and lose trapped air. Warm batter favours fewer, larger, bubbles with thick, soggy walls. Dipping the batter into oil at about 180C abruptly heats the material and causes a popcorn-like reaction: water and air in the cells expand explosively. The cell walls stretch and burst and are rapidly toasted, maintaining their distorted shapes and forming fragile, airy skeletons that give a good tempura texture.  

There are other factors too, for example tempura batter is not supposed to be homogeneous; it is mixed lightly and the slightly clotted texture produces the traditional bubbly appearance of tempura crust. The higher viscosity of the raw batter when cold, helps maintain the necessary lumpiness. It combats convection of the liquid and diffusion of water into cell walls, proteins and starches. In warm batter water diffuses into the amorphous materials, making them soggy. Cooking makes it stodgy and lumpy, rather than forming crisp, microscopic bubbles.

Because, when you whisk up the batter, air dissolves in the batter. Gases are more soluble the colder the solvent, so chilling the batter keeps the dissolved air in solution. Then, when the cold batter is poured/dipped in the hot fat in the pan, the gas becomes instantly much less soluble and tries to bubble out. BUT, the solidifying protein from the batter traps it and makes the cooked batter much less dense (and so lighter) than it would be if it had been left warmer and the air allowed to bubble out of the liquid batter before cooking. My Granny taught me to do this with pancake batter when I was small. A good empirical test is to make pancakes / tempura , cooled and not cooled and eat the products - proof of the pudding and all that.

Sally Ward, Sheffield, UK