Breaking capacity is the stated limit to which a fuse is tested in term of its ability to break a short circuit current.
It is usually measured in Kilo Amps for industrial fuse links. An example of this would be the NH fuse which can safely interrupt up to 120,000A (120KA) at its rated voltage without damage to the fuse link.
The ability of the fuse to interrupt such high short circuits gives the fuse a huge advantage over other protective devices since in most cases other devices can only interrupt much smaller fault currents. In fact, it is often the case that a fuse is required to protect other circuit protective devices due to this.