Transition elements are elements that have partially filled d-orbitals. These elements show variable oxidation state because their valence electrons in two different sets of orbitals, that is (n-1)d and $\mathrm{ns}$. The energy difference between these orbitals is very less, so both the energy levels can be used for bond formation. Thus, transition elements have variable oxidation states.
Related Theory
The presence of unpaired electrons leads to the formation of metal-metal covalent bonds along with the metallic bonds. These strong bonds attribute high melting and boiling points to the elements. The presence of a partially filled $d$-orbital enables the transition elements to have a greater number of unpaired electrons, which in turn increases their ability to form covalent bonds along with metallic bonds.