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.
If the wrong-sized wire is used, or if a wire becomes broken or disconnected, this can cause an accidental short circuit which bypasses the resistance of the component. The current in the wire may become dangerously high and melt the wire or cause a fire.