How do phase transfer catalysts bring anions to organic phase?
Actually, anions are not unstable in organic phases per se. There is no intrinsic physical property of organic phases that would make anions inherently unstable: the main problem is that most heteroatoms in organic solvents are more electronegative than carbon and therefore carry a slightly negative charge which attracts cations more easily than anions.

Most organic phases are best classified as ionophobe, however. They will not like cations or anions in their vicinity. Remember that organic phases are typically mainly hydrocarbons with the odd heteroatom, so they have a low polarisablility. Water, on the other hand, is a highly polarised dipole and able to stabilise both cations and anions easily with the appropriate side of the water molecule.

Now if we add a phase transfer catalyst to a biphasic mixture of water and an organic solvent — typical examples would be $\ce{Bu4N+ I-}$ or crown ethers — then these will readily transfer themselves into the organic phase. In the case of crown ethers, it is because they are basically already like the organic solvent: a hydrocarbon with a few heteroatoms. Crown ethers can, however, piggyback a coordinated cation at practically no extra cost: the cation is directly solvated by the crown ether’s oxygens. In the case of the tetrabutylammonium cation, the charge of the tetravalent nitrogen is somewhat shielded by the four butyl residues which interact much better with other organic residues than with water.

This means that we have a general positive charge in the organic phase due to the crown-ether solvated cations or tetrabutylammonium cations present in there. This in turn draws anions towards the cationic charge due to electrostatic interactions.

We can now also look at some basic principles explaining London interactions: generally, the more disperse the orbitals of a species are, the easier this species can be polarised and thus the better London interactions can be. Cations are species in which there is an overall deficiency of electrons so orbitals are drawn closer to the nuclei. Anions, on the other hand, have a surplus of electrons, meaning that these will repel each other more and orbitals will be more diffuse. Compare neutral atom radii to ionic radii to confirm this. The more diffuse anions will therefore have an easier time forming London interactions with the hydrocarbon solvent than the less diffuse cations.



Thus, in summary, one typically needs the ion pair to cross the phase boundary. An anion might transfer by itself but the cation needs help; thankfully, cations can be transferred easily with crown ethers or tetrabutylammonium compounds because these are more able to interact well with the solvent.