Why did a lack of electrical power generated in Texas recently almost cause fires?

This may be more of an engineering problem…

A recent newspaper article stated that the rolling blackouts instituted during the recent cold snap in Texas narrowly averted parts of their grid catching on fire. Apparently this can occur when demand exceeds supply.

I understand that if there is a "weak link" in a power grid then excessive power through that particular link to supply areas in need might overload it, but the problem seemed to be more general, that whenever demand exceeds supply there could be fires. Why is this the case?