When a person gets fat, do the fat cells multiply or do the fat cells just get bigger

When we consume an excess amount of calories without
utilizing them as an immediate fuel source or "burning them off", the body stores some
in reserve in the form of gylcogen. Glycogen is mainly stored in hepatic and muscle
tissue. The remaining calories are converted to adipose tissue or fat. Adipose tissue is
distributed throughout the body and finds a resting place in all cells. Over time as we
continue to intake excessive calories without utilizing them as a fuel source the
initial adipose tissue cells become larger and the total number of these cells
increase.

When one becomes "fat" the size of the adipose
tissue cells increases and the number of them also increases. Increased  consumption of
calories and especially increased consumption of calories derived from fatty foods is a
major predisposing factor for many chronic illnesses.