Molecular orbital theory gives a good explanation of why metals have free electrons
The best way to explain why metals have "free" electrons requires a trek into the theory of how chemical bonds form. Molecular orbital theory, or, at least, a simple view of it (a full explanation requires some fairly heavy quantum stuff that won't add much to the basic picture) can explain the basic picture and also provide insight into why semiconductors behave the way they do and why insulators, well, insulate.
The first step in getting to a useful intuition involves picturing how small molecules form and how their bonds work. If you start from isolated atoms, the electrons form 'orbitals' of different shapes (this is basic quantum mechanics of electrons). There are plenty of pictures available describing what these look like. But, when atoms come together to form molecules, the simple view of what the clouds of electrons look like gets a lot more complex. If the two atoms form a molecule, they do so because the energy levels of the orbitals in the molecule are lower than those in the isolated atoms for some of the electrons. This is what causes chemical bonding.
The important insight from this picture of bonding is that molecular orbitals don't look like atomic orbitals. When a bond forms, some of the orbitals will fill up with electrons from the isolated atoms depending on the relative energy levels. There may also be other orbitals (some might, were there enough electrons to fill them, form anti-bonding orbitals, weakening the strength of the bond). In some molecules those orbitals might cover a number of atoms (archetypally, in benzene there is a bonding orbital that is shared by all the atoms in the six-membered ring occupied by two electrons and making benzene more stable than the hypothetical hexatriene with three isolated double bonds).
In some solids the picture gets a lot more complicated. In graphite, for example, the bonding orbitals are like benzene but might cover trillions of fused hexagons. What makes the solid hold together is those bonding orbitals but they may cover a very large number of atoms. In metals it is similar.
It is also worth noting that in small molecules you can often get a good idea of the shape of the discrete molecular orbitals, each containing two electrons, when you start dealing with large networks of atoms joined together, the simple, discrete, picture of individual two-electron orbitals becomes pretty useless as there are too many similar ones to make reasonable distinctions.
So solid state chemists and physicists start thinking of the picture as consisting of "bands" of orbitals (or of the energy levels of the orbitals). And this is where we can understand the reason why metals have "free" electrons. When metal atoms come together in a solid, the bonds between the atoms form lower energy orbitals than the isolated atoms. But the orbitals corresponding to the bonds merge into a band of close energies. And those orbitals might not be full of electrons. In metals these orbitals, in effect, form a bond that encompasses the whole crystal of the metal and the electrons can move around with very low barriers to movement because there is plenty of free space in the band.
In insulators, the orbitals bands making up the bonds are completely full and the next set of fillable orbitals are sufficiently higher in energy that electrons are not easily excited into them, so they can't flow around. In semiconductors the same happens, but the next set of orbital bands is close enough to the bands filled with electrons that thermal energy is enough to excite some of them into a fairly empty orbital where they can move around.
Wikipedia give a good picture of the energy levels in different types of solid: . The picture shows both the spread of energy levels in the orbital bands and how many electrons there are versus the available levels. In reality there is a continuum of band widths and gaps between insulators and metals depending on how the energy levels of all the bonding orbitals work out in a particular solid and how many electrons there are to fill them up.
In short, metals appear to have free electrons because the band of bonding orbitals formed when metals atoms come together is wide in energy and not full, making it easy for electrons to move around (in contrast to the band in insulators which is full and far away in energy to other orbitals where the electrons would be free to move).
This is, obviously, a very simple version of reality. But it links the easier theory or chemical bonding and molecular orbitals to the situation in network solids from insulators to metals.