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Fuses are over-current protection devices. They are sensitive to current - thus they are triggered by the amperage that flows through the fuse rather than the voltage (at least, that's the simple explanation).
The voltage rating on the fuse has to do with the approvals on the fuse and the safe operating voltage - smaller fuses generally have smaller voltage ratings because when the fuse "breaks", it needs to be able to quench (stop) the arc that forms and higher voltages will form longer arcs. Higher voltage fuses and those which are intended to stop high fault currents (e.g. HRC) tend to have more than just "air" around the wire (e.g. sand) to help extinguish these arcs.
While a fuse may have a current rating (e.g. 1A), it also has a time component to it as well. Fuses often come in at least two common varieties (fast/quick-blow, slow-blow) and sometimes more exotic varieties (e.g. FF). This is related to the I^2t characteristic curve and is due to the amount of time a given current has to flow in order to melt (destroy) the fusible element. This is why short-term overloads don't always blow the fuse (e.g. inrush current) - but where replacing a slow-blow 1A fuse with a quick-blow 1A fuse can sometimes result in nuisance blowing of fuses - the different fuse time characteristics limit the amount of short-term overload that is permitted by the fuse. Of course, in order to blow a fuse, some power is dissipated over the fuse which in turn implies at least a small voltage necessary to push the necessary amperage across the fusible element.