The basic idea is that if you have a regular periodic lattice then the individual electron wave functions of the atoms can combine together to make electronic wave functions which extend through the entire lattice. That doesn't happen if the material does not have a regular periodic lattice. If the material doesn't have a regular periodic lattice, the individual electron wave functions can still combine with each other, but the resultant wave functions don't extend through the entire lattice of the material.
So if you have a periodic lattice then the electronic wave functions extend through the entire solid, but even so you can still have an insulator, not an electrical conductor. Look at a grain of table salt. It's a nice little cube because its almost a perfect single crystal with a regular cubic periodic lattice. The electronic wave functions in it extend all across the crystal. But yet it doesn't conduct electricity. Why? Because there's another factor to be considered: For an electron to absorb energy and move from one side of the crystal to the other, there has to be an empty available energy level for it to occupy. For table salt, there are no nearby empty energy levels, so salt remains an insulator despite the fact that its electronic wave functions extend thoughout the entire crystal. In other words, there is an "energy gap" as stated in the quote you presented. For crystalline metals, there are available excited energy states, so they can conduct electricity quite well.
A periodic lattice is not required for electrical conduction and indeed there are metallic glasses and metallic liquids. However, since the electrons cannot "sail" through a nice periodic lattice on wave functions which extend though the entire length of the material, the electrical conductivities of metallic glasses and metallic liquids tends to be significantly lower than those of crystalline metals.