Thin wires conduct less easily than thick ones, because there is less room for the electrons to travel through. The energy needed to push current through a resistance is transformed into heat. This can be useful, for example in the very thin filament of a light bulb, which glows white hot.
Depending on the fuse design, it is sometimes possible to tell whether it has blown by holding it up against a light; a break in the wire inside may be visible. Another clue is blackening of the glass cover. If there is no visible sign, check by fitting another fuse of the same rating; if that cures the trouble, then the fuse was to blame.