I'd never even heard of this device until I read your post. Here's what I found on the net though:
A sample of the spirit to be tested is poured into the test jar, and stirred with the thermometer. The Hydrometer is inserted and allowed to float. The temperature reading and hydrometer reading are taken and the results compared with the chart which will provide the strength in % alcohol by volume. The chart is suitable for all spirits such as gin, vodka and whisky whose alcohol content is not obscured by a high level of sugar, caramel or other solids.
It sounds like the hydrometer measures liquid density; the deeper the hydrometer floats in the liquid, the less dense is the liquid and therefore the more alcohol is in it. As density varies with temperature, you need to measure that too.
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