Looking at it from the prototype (real trains) -- 1/48 means 48 times smaller than the real thing. It's in feet. 1 foot on your layout equals 48 feet in the real world.
Looking at it from the model (O scale trains) -- 1/4 means that one foot in the real world occupies 1/4" on your layout. It's in inches.
The arithmetic (tying both together) -- 1/48 x 12 inches per foot = 1/4"
O Scale is proportioned 1:48, expressed as 1 to 48. The scale is 1/4 inch per foot. There are 4 x 12 1/4 inches in a foot, 4x12 is 48. You probably should not see it expressed as 1/48, but 1:48.
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