I don't think I could improve on section 7 of Physics Reports 164 (1988) pp 217-314, Nonlinear realization and hidden local symmetries, by Masako Bando, Taichiro Kugo, Koichi Yamawaki, (beyond attributing the chiral model to the magnificent F Gürsey (1960), a decade before the types these reference...); and certainly not the connections to AdS/CFT and Seiberg duality that modern students of Sakurai's Vector Dominance scurry to at the drop of a hat.
I could just sketch a geeky cartoon of what the effort is all about, in the simplest $SU(2)_L\times SU(2)_R/SU(2)_V$ model, on the unitary $U=\exp(i\vec \pi \cdot \vec \tau /f_\pi)$.
The corresponding right-invariant, left-invariant, unbroken vector, and broken axial isotriplet symmetry conserved currents are
$$
\vec L _\mu = i f_\pi^2 \operatorname{Tr} \frac{\vec \tau}{2} U\partial_\mu U^\dagger = f_\pi \partial_\mu \vec \pi +... ,\\
\vec R_\mu = i f_\pi^2 \operatorname{Tr} \frac{\vec \tau}{2} U^\dagger \partial_\mu U =- f_\pi \partial_\mu \vec \pi +... ,\\
\vec V_\mu = i f_\pi^2 \operatorname{Tr} \frac{\vec \tau}{2}\left (U\partial_\mu U^\dagger+ U^\dagger \partial_\mu U \right ) = -2~\vec \pi \times \partial_\mu \vec \pi +... ,\\
\vec A _\mu = i f_\pi^2 \operatorname{Tr} \frac{\vec \tau}{2} \left (U^\dagger \partial_\mu U -U\partial_\mu U^\dagger \right ) = -2f_\pi \partial_\mu \vec \pi +... ~~.
$$
The s.broken conserved currents start linear, the unbroken ones (V) bilinear. So, the axial currents are some sort of pion spray-gun aimed at the vacuum, but the unbroken vector ones have the exact quantum numbers of the ρ isotriplet ... from time immemorial
people have been modeling this lightest vector as a "composite" of pions in the form of $V_\mu$... how can you avoid it?
Significantly, if you skip the Ws and Z and just gauge the chiral action through electromagnetism, so couple a photon to $V_\mu^3$, and blink, it very much looks like the action has a $\gamma \rho $ mixing term , so very useful in modeling low energy interactions of photons with hadrons. It looks like the photon does not couple to the relevant hadron (proton...) through its charge, but, rather, it "converts" to a ρ which then talks to the hadron in its inimitable strong way.
This is called VDM, linked above, and every model of ρ in terms of pions must replicate the phenomenological successes of this description.
The task for such modelers, then, is to somehow invent a mechanism of producing ρ s as pion composites. There had been a half a dozen years of such efforts, starting in the late 70s, until BKY, above, came with a geometric vision/scenario which I won't replicated, based on the Maurer-Cartan form geometry of chiral models.
Basically, a hidden gauged symmetry like V is posited and SBroken, whose vectors are then higgsed through a gauge mechanism of extra σ s, to produce the massive ρ s as pion composites, exactly through an on-shell identification with the vector currents V, above.
So the coset space's H (isospin V) survives, but its shadow ("hidden") double is SBroken and Higgsed. The theory apparently produces all kinds of phenomenological successes of the past, KSFR relations and the like, but I lack critical appreciation of how "easy" or not such are, to avoid the imprecation of "likely in such a tight structure", anyway.
All in all, it is a justifiably very influential idea, and should not be ignored on the trail...
