First off, it's a liquid probably because the hydrocarbon chains can't effectively pack to form crystallites and solidify. If you considered just the branches, as if they were octane or decane, these compounds are also liquids at room temperature. You need longer hydrocarbon branches to start getting solids like waxes.
As for it's utility as a phase transfer catalyst, that may be related to increased solubility in organic phases. Tetramethylammonium chloride only has those four little methyl groups to try and mask the ionic site in nonpolar media, while Aliquat 336 has much longer hydrocarbon chains. This means it's solubility in the organic phase is going to be better than tetramethylammonium chloride's.
I suppose the longer hydrocarbon chains could also kind of anchor the Aliquat 336 to the interface, preventing them from diffusing too far into the polar phase to get back to the nonpolar phase efficiently, while still being long enough chains to allow the ionic site into the polar phase to grab a reactant.