What causes straight fatty molecules to form a lattice (e.g. saturated fats)?
Nature abhors a vacuum, and your sticks aren't sticky. 

When you have a jumble of sticks, there's lots of voids and spaces between them - the pile of sticks takes up much more room than the bundle. For wooden sticks, this is no problem, as air can easily fill in the voids between them. With saturated fats, this isn't possible. There really isn't anything that can fill the voids. The only way to fill them is to line up the chains in an ordered lattice.

Also, when the chains are in an ordered lattice they're making a large number of van der Waals interactions between them. Because the chains are straight, the best way to maximize these favorable attractive interactions is to pack the chains together in an ordered array. The van der Waals forces are comparatively small, but over a long chain it adds up. (I haven't ever seen this done, but I'm guessing that if you coated your wooden sticks with something sticky like honey, and shook them around a bit, your wooden sticks would also want to line up with each other and stick together in an ordered array.) 

Temperature, of course, can overcome both effects. The additional kinetic energy in the molecules allows them to overcome the attractive interactions between the molecules, and higher temperatures mean that the molecules are moving faster, pushing away other molecules and effectively filling in the voids to some extent. When you get to a high enough temperature, the fat melts.

In unsaturated fats, the chains are kinked. This means that it's much harder to line them up to form all those van der Waals interactions, and there are intrinsically voids in even the most ideal packing. Thus it's much easier to break those interactions and add the additional voids needed for a liquid state, so the solid to liquid transition happens at a lower temperature.

One thing to keep in mind is that things like solid/liquid transitions are not absolute things. Where the solid to liquid phase transition happens is based on the relative stability of the solid versus the liquid. It isn't that the saturated fat can form a solid and the unsaturated fat can't, it's just that the solid saturated fat is more stabilize relative to the liquid saturated fat, than the solid unsaturated fat is to the liquid unsaturated fat. As the change in stability is much greater for saturated fat, it takes a higher temperature to melt the saturated fat than it does the unsaturated one.