Recently, I learn about bonding and my teacher only focus on ionic and covelant bond, not metallic bond. So, I go on google search but I have no idea of what is it? How electrons can suddenly appear to form the bond? How can the metal highest energy level overlap if it need to obey the octet rule
Metallic bonds depend on the Pauli exclusion principle to cause a single energy level to be separated into a sea of very closely spaced levels.
Consider metallic Ca, with its s2 filled with two electrons of opposing spin state. You cannot have another s2 electron in the same atom. Now, add a second Ca atom that can freely interchange electrons with the first (e.g. metallic conductivity). Quantum mechanically, the electrons are indistinguishable, and an electron of the first atom might be found on the second one, so the combination acts as a "super-atom" with two nuclei but a single electron cloud. To allow this without violating the Pauli exclusion principle, one s2 level "moves up" and the other "moves down", splitting the level into two bands.
As more atoms are added, the Fermi "sea" becomes populated to the extent that the energy levels merge into a continuum.