The hydrocarbon chains are, of course, in constant motion in the liquid, but it is possible for them to lie tidily when the substance solidifies. If the chains in one molecule can lie tidily, that means that neighboring molecules can get close. That increases the attractions between one molecule and its neighbors and so increases the melting point.
Unsaturated fats and oils have at least one carbon-carbon double bond in at least one chain. Rotation about a carbon-carbon double bond is constrained, locking a permanent kink into the chain. This makes packing molecules close together more difficult. If the chains cannot pack well, the van der Waals forces will be less effective. This effect is much stronger for molecules in which the hydrocarbon chains at either end of the double bond are arranged cis- to each other, as shown in the figure below: