What is the intuition behind this and the distinction Skyrmion of
pseudo-scalar mesons and vector mesons?
I'm not sure, but the authors may just state the problem of non-stability of skyrmions in "minimal" chiral perturbation theory.
The skyrmion itself is the non-point-like object characterized by some radius $r$. It can be shown by using the simple scaling arguments that in chiral perturbation theory with the lagrangian
$$
L = f_{\pi}^{2}\text{tr}\left[|\partial_{\mu}U|^{2}\right]
$$
the skyrmions are unstable, i.e., the radius $r$ tends to zero even if initially it is non-zero. Therefore the baryons can't be described by the skyrmion solutions in this theory.
There are two ways to solve this problem. The first one is to assume the non-minimal lagrangian
$$
L = f_{\pi}^{2}\text{tr}\left[|\partial_{\mu}U|^{2}\right] + a \text{Tr}[|\partial_{\mu}U|^{4}]+...
$$
Within this theory, the skyrmion solutions are stable.
The second one is to introduce background massive vector fields $V_{\mu}$ minimally coupled to $U$. The lagrangian now is
$$
L = f_{\pi}^{2}\text{tr}[|D_{\mu}U|^{2}] - \frac{1}{4}V_{\mu\nu}V^{\mu\nu}, \quad D_{\mu} = \partial_{\mu} - igV_{\mu}
$$
These vector fields are identified as the vector mesons. This approach sometimes is called "$\rho$-stabilized skyrmions", by the name of the lowest mass vector meson, the $\rho$-meson.
If you need, I can add the clarifying details in the answer.
And their relations to baryons or other hadrons?
Starting from the QCD (with the $SU_{L}(3)\times SU_{R}(3)$ global symmetry) and constructing the chiral perturbation theory, the skyrmion can be shown to have half-integer spin (1/2, 3/2,...), non-zero mass and integer baryon number (in dependence on the corresponding topological number). Next, the skyrmion is non-point like object with finite radius. Finally, starting from the skyrmion solution $U_{\text{skyrmion}}$, introducing the perturbations around this solution (which are parametrized in terms of the pions), it can be shown that the skyrmion-pion vertices are similar to the couplings of the nucleons to the pions.
