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Mitogens
Mitogens are the chemical signals that cause cells to divide. Until the mitogenic signal is given, cells remain intact and therefore larger. Typically mitogens are stimulated by other cells nearby. An abundance of nutrients to feed both the dividing cell and the "new" cell is also needed.
Extracellular Factors
The growth of cells is heavily influenced by factors outside the cell itself. Neighboring cells exchange signals and messages that instruct each other to grow and increase in size. In turn, they exchange proteins and other macomolecules. In one example PI 3-kinase, a cellular chemical signal, stimulates protein synthesis in nearby cells, which cause it to increase in size.
Geometry
Basic geometric principles are also at work in determining cell size, in particular the relationship between volume and surface area. As surface area increases arithmetically, volume increases geometrically; eventually the volume increase outpaces the surface area increase, and the cell divides. Some cells have adapted to "overcome" this limitation. Nerve and muscle cells, for example, are long and thin, which gives them greater surface area.
Cellular Function
The specific function of a cell also influences its size. This is known in biology as "differentiation." For example, red blood cells are, in cellular terms, quite large, while the rod cells in our eyes are especially small. Cells are naturally limited in size by their ability to receive nutrients and expel waste. If they grow too big, they cease being able to perform such functions.