Wikipedia told me that breaking capacity is the maximum current that can safely be interrupted by the fuse. I don’t understand why, if a small current can blow the fuse, a bigger current can’t. If the current which is bigger than breaking capacity will cause arc, why a small current with the same voltage won’t?
To elaborate a bit on the answer by Neil_UK...
At a modest overload, the fuse wire will melt at its weakest point, and break the current.
At a larger overload, an arc will form across the ends of the broken wire. This arc will persist until more wire has melted and the gap is too long to sustain the arc.
At a massive overload, the wire will vaporize. The metal vapor will support an arc running the entire length of the fuse. This arc will persist until either something else breaks the current, or the fuse goes bang.
High current fuses are often sand-filled to help quench the arc, and have hard ceramic bodies, rather than glass, to resist the explosion.
Addendum, after some comments on the question and the other answer.
Ideally, the fuse should be rated to break the maximum prospective fault current for the circuit it's protecting. That is, the maximum current that could flow if you put a dead short across the output of the fuse, taking into account the size of the supply transformer and all of the cabling back to that transformer.
Sometimes that isn't practical, and you have to rely on upstream fuses blowing in the most extreme short-circuit cases. That can be acceptable if you know that the upstream fuse will blow before the downstream one fails catastrophically.
I don’t understand that if a small current can blow the fuse, why a bigger current can’t.
A bigger current will indeed melt the fuse wire. The question is, what happens after that?
If the fuse is too small, so the current it's trying to interrupt is above its maximum current rating, then the arc may fail to quench, and continue to conduct for a long time after it should have 'blown'. Fuses often contain materials to cool and extinguish the arc, sand for instance. If more energy is dumped into it than it's designed to quench, then it won't quench.
In a more extreme failure mode, the fuse may physically explode.
Say, we have a 13 A fuse in a plug.
At <13 A of continuous current - fuse won't blow; this is the range of current that the fuse can handle safely for an indefinite amount of time
At 13-20 A of continuous overload current - fuse won't blow but surrounding parts in the plug may overheat
At 22 A of continuous overload current - fuse will blow within minutes to hours; a 13 A fuse will blow at an overload current about 1.6× its rated current
At 50 A of continuous overload current - fuse will blow within 0.1-20 seconds
At 400 A of fault current - fuse will blow in <0.04 seconds
At 3000 A of fault current - fuse will blow instantaneously; a 13 A fuse can blow safely up to 6000 A, otherwise known as its breaking capacity
At >6000 A of fault current - fuse may explode or cause a dangerous electric arc