There's still a voltage, it just doesn't register on the meter - it's a measurement anomaly.
Think of it in terms of Isaac Asimov's water analogy. Voltage is the pressure of the water, which is supplied by the source. Current is the amount of water being moved. Resistance is the size of the pipe. If pressure exists between the input and output of the source, water will flow, so you'll measure a current as long as the connection exists. To actually have no flow you need to turn off the source - i.e. actually break the connection to the battery, turn off the power supply's switch, etc. Then there's no pressure on the water in the pipe, If there's pressure across the pipe, water will flow.
When a short circuit exists it's the same as an infinitely (in theory) large water pipe, so your meter, which has limited sensitivity, doesn't detect any pressure difference. However since there's no such thing as an infinitely large pipe there's still flow, but your meter can't detect it because the pressure difference across the pipe is close enough to the internal pressure difference of the source that the meter, which detects a pressure differential between the two, can't see it. But there's still a lot of water moving.