How inductance of a coil changes when there is a metal object nearby?
What actually happens when a metallic object is put under a coil that
is meant to be used as a detection coil for metal detector?

If the metal is (say) iron then there are two opposing effects. Iron has a magnetic permeability much greater than 1 and this will tend to increase the inductance of the search coil but, iron is also a conductor and, it will act as a shorted turn and reduce the inductance of the search coil. Somewhere between the two is the answer and this depends on: -

Operating frequency
Relative dimensions of the piece of iron (i.e. shape)
Size

Incidentally some grades of stainless steel are very difficult to find using a metal detector because the two effects cancel out. I discovered this when designing metal detectors for food and pharmaceutical lines.
For other (non-ferromagnetic) metals/conductors their presence reduces the inductance of the search coil.