How is this expression for a GHZ state obtained in the nature paper by Pan et al. (2000)?

First, in order to express $|H'rangle$, $|V'rangle$, $|Rrangle$ and $|Lrangle$ in terms of $|Hrangle$ and $|Vrangle$, add and subtract the pair of equations $(2)$ and add and subtract the pair $(3)$, to get

$$ |H'rangle + |V'rangle = sqrt2|Hrangle |H'rangle - |V'rangle = sqrt2|Vrangle |Rrangle + |Lrangle = sqrt2|Hrangle |Rrangle - |Lrangle = isqrt2|Vrangle $$

which means that

$$ |Hrangle = frac{1}{sqrt{2}}left(|H'rangle + |V'rangleright) = frac{1}{sqrt{2}}left(|Rrangle + |Lrangleright) |Vrangle = frac{1}{sqrt{2}}left(|H'rangle - |V'rangleright) = frac{1}{isqrt{2}}left(|Rrangle - |Lrangleright). $$

Substituting the last equations into $(1)$ according to the choices of polarization in text, we obtain

$$ begin{align} |Psirangle &= frac{|Hrangle|Hrangle|Hrangle + |Vrangle|Vrangle|Vrangle}{sqrt{2}} & = frac{(|Rrangle + |Lrangle)(|Rrangle + |Lrangle)(|H'rangle + |V'rangle)-(|Rrangle - |Lrangle)(|Rrangle - |Lrangle)(|H'rangle - |V'rangle)}{4}. end{align} $$

Notice that the only terms that do not cancel are those with odd number of kets in ${|Lrangle,|V'rangle}$, i.e.

$$ |Psirangle = frac{1}{2}left(|Lrangle|Rrangle|H'rangle + |Rrangle|Lrangle|H'rangle + |Rrangle|Rrangle|V'rangle + |Lrangle|Lrangle|V'rangleright) $$

in agreement with $(4)$.