Does "stage" tell you how much water is flowing in a stream?

Not directly. You cannot say that because a stream rises (doubles) from a 10-foot stage to a 20-foot stage that the amount of flowing water also doubles. Think of a cereal bowl with a rounded bottom. Pour one inch of milk in it. It doesn't take much milk to make it up to the one inch level because the bowl is at its narrowest near the bottom. Now, pour in milk until it is two inches deep -- it takes a lot more milk than it did to fill the first inch because the bowl gets wider as you go up. The same thing happens in a stream; the stream banks will generally be narrower at the bottom and widen as you go up the bank. So the amount of water flowing in a stream might double when the stage rises from 1 to 2 feet of stage, but then it might quadruple when it goes from 3 to 4 feet. 

To find out how much water is flowing in a stream or river, USGS personnel must physically measure how much water is flowing (“discharge”). That volume is usually expressed in cubic feet per second. This is often accomplished by standing in the creek and measuring the depth and water speed at many places across the creek. By doing this many, many times and at many different stream stages, we can develop a relation between stream stage and discharge called a rating curve.

Learn more:

• How does a USGS streamgage work?
• How Streamflow is Measured