How does the body figure out which side is left and which side is right? I can think of an easy way to break symmetry but it seems harder to consistently know which side is which.
Short answer: at the single cell stage there's a cilium that, spins in a single direction and establishes chirality. People with immotile cilia thus have a 50/50 chance of getting SI, because the developing body chooses randomly.
I thought it was multiple cilia, wafting fluid in a current around a cavity, a bit later than the single cell stage but still very early?
Either way, the reason a cilium spins in one direction rather than another is that the motor is a molecular machine built out of amino acids, and amino acids are themselves chiral. If you made a person out of amino acids with the opposite chirality, their cilia would go the other way, and their body would be flipped. It's a lovely example of magnifying something up from the molecular to the macroscopic.
The motion of the cilia is not directly coupled to the chirality of the amino acids, you could probably make a cilium that obligatorily moved in the other direction with a handful of mutations.