When reading across a fuse the voltage will take the less resistant path and go through the fuse, not your meter which has resistance built in. When the fuse is blown, there is no voltage flowing though it. Connecting your meter, you create a new path for the voltage to take, this getting a reading.
Edit: the potential is still on the one side of the fuse. You are just creating a new bridge. Also that should be the only voltage you see unless there is a transformer or the like.
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