Parasitic Inductance
Generally, an inductance (parasitic or otherwise) indicates energy being stored in (and recovered from) the magnetic field.

Any current through a wire produces a magnetic field around the wire. But that magnetic field stores energy. Therefore you can't produce the current instantaneously (unless you can deliver infinite power). And if you want to shut the current off, you need to absorb that energy somewhere else. 

With a bit more math to put numbers to things (how much current produces how much magnetic field, storing how much energy) you have your inductance.