why transistion metal exhibit variable oxidation state?

The reasons for the variable oxidation states of the transition metals :

(1) The valence electrons involve (n-1)d and ns electrons, because their orbitals have close energy levels.

(2) Differences between successive ionization enthalpies are small. Therefore, difference numbers of electrons may be lost from the transition metal atoms, leading to cations carrying different positive charges, e.g. Fe²⁺ and Fe³⁺.

(3) Vacant valence (n-1)d orbitals are available. Therefore, valence electron(s) in lower energy level can promote to vacant orbitals in higher energy level to make more incompletely filled valence orbitals, and thus more covalent bonds can be formed.