What is the mechanism of folic acid deficiency and neural tube defects?
You are having difficulties finding the mechanism because it is as of yet unknown. 

Now of course comes the hard part where I need to prove that statement. Often, it's just apparent by papers not saying anything about the mechanism, even though you'd expect then to include a brief section if it was known, for example Folic acid supplementation for pregnant women and those planning pregnancy: 2015 update or the various Cochrane reviews on the topic of how much folic acid supplementation is optimal, or the WHO and CDC documents on the topic 

Here's a few relatively recent ones that state it outright:

Folic Acid Supplementation and Pregnancy: More Than Just Neural Tube Defect Prevention, 2011


  The mechanism by which folic acid prevents structural anomalies in the fetus is not known, but may involve the regulation of homocysteine metabolism


Folic acid supplementation in pregnancy and implications in health and disease, 2014


  as FA deficiency may be a risk factor for NTDs additional studies will be required to determine the mechanistic role of the FA pathway in the onset of neural tube defects


FA = Folic Acid, NTD = Neural Tube Defects 

Neural Tube Defects, Folic Acid and Methylation, 2013


  To date, animal studies have not provided 
  sufficient information to establish the metabolic and/or genomic mechanism(s) underlying human folic acid responsiveness in NTDs.


There are some putative mechanisms to do with folic acids effects on the homocysteine metabolism and its effects on methylation. I can't really tell how well-established those are yet, but maybe this can be a starting point for you.