In this kind of context, many will regard neither as being grammatically singular, and therefore requiring a singular verb, as in your example. Not all authorities agree with this analysis, however, calling in aid the concept of ‘notional’ or ‘proximity’ agreement, which would allow a plural verb after neither of. As ‘The Cambridge Guide to English Usage' reports, plural agreement ‘is strongly associated with spoken discourse. In the British National Corpus it’s used in 75% of all instances from transcribed speech, but only about 20% of instances from written texts.’
It may be this preponderance of plural agreement in speech that has led you to expect understand rather than understands here. I imagine that the example you give is indeed speech, rather than part of the narrative of the book. If it is, then the author may have had some characterisation purpose in having the speaker use a form that is untypical.