To understand this, it is important to have a look at a graph of current versus time to fuse of a 13 Amp fuse.
The left hand pair of curves is for a 3 Amp fuse and the right hand pair is for a 13 Amp fuse. The pair of curves either side of the text "13 A" represents the tolerance for the fuse.
For example, a 13 Amp fuse can take 20 Amp indefinitely without blowing. ( See the vertical line representing 20 Amps is parallel to the curve).
If 100 Amps were to flow through a 13 Amp fuse, it will blow someweher between 0.01 and 0.3 of a second.
If 50 Amps were to flow through the 13 Amp fuse, it will take somewhere between 0.1 and 11 seconds to blow.
So fuses are there to protect against large overcurrents due to faults and short circuits. Overloading a fuse by a small amount can lead to dangerous overheating as can be seen below.
An obvious follow up questions is "Why don't manufacturers design a 13 Amp fuse to blow at 13 Amp?"
Well, 13 Amp fuses are designed to pass 13 Amps continously without blowing. Fuses blow when they heat up. It is virtually impossible to design a fuse that does not blow at 13 Amp but definitley blows at say 14 Amps due to the tolerances involved. The graph above is a presentation of what can be achived practically.
It is also interesting to note that this important topic is not dealt with at all in the IET Code of Practice on PAT Testing.
In summary, a good understanding of how fuses work is useful in preventing fires in the workplace.