Since matter can't be created or destroyed, only transferred or transformed, what is the scientific reason on why cells in living organisms dividing can lead to them growing or getting bigger.
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It's why you need to eat, and why you, literally, are what you eat. Your body takes the matter in food and uses it to build itself up. The same is true of energy (which, like matter, can neither be created nor destroyed). Plants use energy from the sun and store it in molecules like glucose and other sugars. Animals harness this energy when they eat the plants.
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Organisms integrate matter and the energy necessary to carry out the integration. The process is fundamental to life. (If you don't eat, you die.)
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Simple: they eat. Living organisms capture matter from their surroundings and integrate it to their own structure.