Tom gave up the brush with reluctance in his face, but alacrity in his heart. And while the late steamer Big Missouri worked and sweated in the sun, the retired artist sat on a barrel in the shade close by…. How does Tom's allowing Ben to whitewash the fence move the plot forward?
It causes Ben to encourage other boys to help whitewash the fence.

It leads to Tom leaving to go swimming.

It leads to other boys giving Tom things to let them whitewash the fence.

It causes Ben to forget about eating his apple.