I am not sure I would say they are generally pyrophoric. Ferrocene is certainly not and there are many, many complexes described that are not pyrophoric. Now, with respect to many compounds such as lithium, sodium, magnesium , aluminum alkyls the polarization of the bonds make reactions with oxygen ver facile and the liberation of heat and the presence of oxygen is sufficient to combust the organic fragment. There are a number of organometallic texts out there that you should look up and consult.
This can be a potentially heated (no pun intended) debate. While organometallics like Zeises salt, Ferrocene, Cp*Ir(III) are quite stable to air (moisture and O2), metal alkyls aren't.
I am assuming that by the phrase 'organometallics being pyrophoric' the author is referring to compounds like alkyl or aryl lithiums and magnesiums etc.
One way to crudely generalise is to think of this as follows:
C-H (almost covalent) goes to C-M (where C is negative, M is positive). This transformation is highly endothermic.
When exposed to a proton such as with an acid HX, the C-H bond and MX (lattice energy) formation leads to release of huge amounts of energy.
Sometimes consideration of pKa also helps:
pKa of alkanes are ca. 50, while that of Cp is 16. Hence metal-alkyls are generally unstable while Cp complexes are generally stable. ['Generally' is the most critical word here]