My answer is very similar to MaxW's answer, but maybe it can still provide a
bit of insight.
Consider methane as our current molecule:
1 carbon atom reacts with 4 hydrogen atoms to form 1 methane molecule.
Most/Many textbooks explain this by hybridization of three p orbitals
and one s orbital to form four energetically equivalent (degenerate) sp3
orbitals. This is the valence bond model.
"[Carbon] has only two singly occupied orbitals available for bonding."
The author suggests that since only two electrons are available carbon should
form two bonds. This would leave the last carbon p
orbital completely empty, though, and, tbh, this would to me be completely
counterintuitive. To say it in a sloppy way: Chemical bonds/orbitals are
all about being either completly full or completly empty (half-filled is also
possible sometimes). Leaving one of the three p orbitals completly empty is
absolutely not what any chemist would expect in such a situation. To suggest
that carbon would only form two bonds is actually the really counterintuitive
thing here (at least for me). It would even make more sense that carbon loses
its two p electrons in order to have noble gas configuration, but ok... let's
leave that aside
"Why don't the 4 added electrons just go into the px and py and some new pz?"
The thing is that every bonding orbital can contain maximum two electrons. A bond between two atoms is only established, if every atom provides one electron each. If one carbon p orbital is empty then it would not make sense to put in two electrons from hydrogen. If two hydrogen electrons are put in the same bonding orbital it only means that the two hydrogen atoms are now bonded and form an H2 molecule.
So since every electron from the 4 hydrogen atoms needs an electron from
carbon, the best way is to take the two electrons from the 2s
shell. In that way carbon can provide its two 2s electrons for two bonds with
two hydrogen atoms and the 2 p electrons can bind with another two hydrogen
atoms. This will give four bonding orbitals, which are all completely full.