What caused a fall in atmospheric oxygen about 1.9 billion years ago?
The answer might lie in aerobic respiration. This is of course only one of the many possible explanations but mitochondria (and therefore $\ce{O2}$-breathing eukaryotes) are thought to have evolved circa 2.3 to 1.8 Ga (see for instance Hedges et al., 2001; 2004; 2006). This loose age bracket however is based primarily on molecular data with very few fossil evidence (as I mentioned in an earlier answer of mine, the Proterozoic fossil record of eukaryotes is almost inexistant).
An earlier apparition of aerobic respiration is unlikely (e. g. Knoll &amp; Holland, 1995) as a certain level of $p_{\ce{O2}}$ needed to be reached for this metabolism to be possible.
If aerobic eukaryotes expanded rapidly they might have constituted a strong enough sink of oxygen, which could be responsible for the drop observed by Frei et al., 2009.