Generally speaking, the "classical" explanation of normal anion gap metabolic acidosis has always been somewhat unsatisfying. The textbooks have called it "hyperchloraemic" acidosis, and have discussed it in terms of bicarbonate loss as the chief problem. In this fashion, the acidosis which accompanied normal saline administration was described as "dilutional". The main disturbance was thought to be the consequence of a stable amount of bicarbonate becoming dispersed in a larger amount of fluid, this process giving rise to a lower bicarbonate concentration and thus acidosis. 

The quantitative approach to acid-base physiology explains the change in pH by using a different mechanism. The bicarbonate is viewed as a dependent variable, a plaything of the strong ions - and the chief determinant of its concentration is the "strong ion difference", the potential bicarbonate space generated by the difference in sodium and chloride concentration in the body fluids. Proponents of this approach offer robust arguments regarding its validity; opponents respond with  well-reasoned objections  which are definitely worth reading. Without getting involved in the debate and without any pretence of biochemical expertise, the author confesses that he has become attracted to the intoxicating mathematical integrity of the Stewart interpretation. Therefore, in the majority of these chapters, the Stewart explanation is offered as the dominant one, rather than as an interesting alternative. This is done intentionally, acknowledging that it has several major limitations and that it may not contribute very much diagnostically. Apart from personal preference, the argument for this trend stems from the clinical interchangeability of the two explanations, as well as the convenience of using a method which appears to be favoured by our  College of Intensive Care Medicine, or at least by some of its most productive and respected fellows

Anyway. Who doesn’t like Gamblegrams?

Using the strong ion difference to explain normal anion gap metabolic acidosis

gamblegram of the strong ion difference

In general, Stewart’s approach is fascinating, but to elaborate on it would require much more space than I am willing to allocate at present. In any case, others have done a better job of it than I could ever hope.

In summary, it demonstrates that changes to the difference between chloride and sodium are responsible for the decrease in bicarbonate and the resulting increase in H+.

Lets gamblegram an increase in chloride.

Gamblegram of decreased strong ion difference

So, if the chloride increases, there is no way to maintain electroneutrality EXCEPT by decreasing bicarbonate; certainly the albumin and the other anions aren’t going anywhere. The principle of electroneutrality demands a decrease in bicarbonate.

The bicarbonate concentration participates as the numerator in the Henderson-Hasselbalch equation; a decrease in bicarbonate results in a decrease in pH (or increase in H+, if you will) if the CO2denominator remains the same.  

So where does the bicarbonate go? Well, it performs as a buffer, as it always would.

Conventionally, any excess H+ will combine with HCO3- to briefly exist as carbonic acid (H2CO3) before being chewed up by carbonic anhydrase and turned into H2O and CO2. So these two equations can be viewed as being in equilibrium. As the strong ion difference decreases, a decrease in bicarbonate results in a decrease in pH, which produces plenty of H+ ions for the HCO3- to bind with, thus removing some of the HCO3- and the H+ from the electroneutrality calculations. This equilibrium comes to rest at a lower pH, provided the CO2 concentration remains the same. 

References

So, you have the insatiable lust for mathematically coherent explanations of acid base balance?
Well! Have I got links for you.

AcidBase.org is the official home of Stewart’s hypothesis (and a monument to Stewart).

Kerry Brandis has predigested this difficult concept for people like me. I direct the discerning reader to this page (and the next few pages).

One other article does NAGMA much better justice than my ranting ever could.

Story DA. Hyperchloraemic acidosis: another misnomer? Crit Care Resusc. 2004 Sep;6(3):188-92.

LifeInTheFastLane does a beautiful job of summarising NAGMA, and provides helpful mnemonics (sadly, none of them are dirty).

Kirchner, Karl, and Bruno Schneeweiss. "A critique of Stewart’s approach: the chemical mechanism of dilutional acidosis." Applied Physiology in Intensive Care Medicine 1: Physiological Notes-Technical Notes-Seminal Studies in Intensive Care 1 (2013): 71.

Oh, Man S., and Hugh J. Carroll. "The anion gap." New England Journal of Medicine 297.15 (1977): 814-817.

Goodkin, DAVID A., RASIB M. Raja, and A. L. A. N. Saven. "Dilutional acidosis." Southern medical journal 83.3 (1990): 354-355.

Constable, Peter D. "Hyperchloremic acidosis: the classic example of strong ion acidosis." Anesthesia & Analgesia 96.4 (2003): 919-922.

Masevicius, Fabio D., and Arnaldo Dubin. "Has Stewart approach improved our ability to diagnose acid-base disorders in critically ill patients?." World journal of critical care medicine 4.1 (2015): 62.

Adrogué, Horacio J., et al. "Assessing acid–base disorders." Kidney international 76.12 (2009): 1239-1247.