I guess dipole interactions have the biggest influence here. They somehow destabilize the reversion of the oxaposhpetane to the betaine and prevents formation of the more stable E isomer but I am not sure.
As you hopefully realise, the stereochemical outcome of the HWE reaction (and variations) is due to the relative orientation of the R-groups in the phosphaoxetane intermediate.
If the R groups are syn as in 5A then the (Z)-olefin is formed, with an anti arrangement 5B favouring the (E)-olefin as the product.
Source: Modern Carbonyl Olefination, Wiley, 2004
The pertinent question when looking at the selectivity is why do we get dominant elimination from the syn phosphaoxetane (leading to the correct olefin), rather than dominant elimination from what is arguably the sterically most favourable anti phosphaoxetane (in which the R groups are pointing away from one another).
As it turns out, the initial 'aldol' addition into the aldehyde is rate determining, with the anti addition being favoured over the syn addition on steric grounds.
Source: Stereoselective alkene synthesis, Springer-Verlag, 2012 (a really useful book, FYI)
Once the aldol addition has taken place, the subsequent steps are rapid due to the high electrophilicity of the Still-Gennari phosphonate (with the additional electron withdrawing groups).
This means that the entire process is essentially controlled by the initial addition to give a syn or an anti adduct. Once formed, the adducts are sufficiently short lived that despite the cis phosphaoxetane being less favoured due to the clash between R-groups, the (Z) olefin is formed via quick and irreversible elimination.
The HWE therefore is (E)-selective, in part, due to the fact that elimination is less rapid and as such there is equilibration to the more favourable phosphaoxetane from which elimination may occur.
I am not convinced by the transitions states given in "Stereoselective Alkene Synthesis" because they rely on chelation to restrict the conformation of the phosphonate anion AND the orientation of the aldehyde.
And yet, the reaction conditions are KHMDS/18-crown-6 which are clearly designed to eliminate the possibility of coordination to the potassium cation and maximise the rate of addition of the "naked" alkoxide anion to the electron deficient P.
My own suggestion for the intrinsic "anti" (in this example) selectivity of the fluorophosphonate anion is that it simply mirrors the intrinsic selectivity of the unstabilised Wittig ylid and Horner-Wittig (Ph2PO) stabilised anions both of which produce Z-alkenes stereoselectively.