Why do roots grow downwards and stems grow upwards?

'Gravitropism' is the reason for roots growing downwards and the stem growing upwards. Gravitropism is simply a plant's growth movement in response to gravity.

The roots of a plant show 'positive gravitropism' as the roots respond to the stimulus of gravity and grow in the direction of gravity (downwards). This phenomenon is called 'geotropism'. 

The term Geotropism is derived from two words, "geo" which means earth or ground, and "tropism" which means a plant movement triggered by a stimulus.

While the stem displays negative gravitropism as the shoots respond to the stimulus of light and grow in the direction of light (upwards). This phenomenon is called 'phototropism'.