@comicalmayhem, you say you’re not asking about single-cell to multicellular, but that’s a huge part of the answer. As for the “how,” we can look at some extent organisms that seem to skirt the boundary between single-cell and multicellular. For example, slime molds that act as single cells, except when conditions pressure them to act “socially” and join together into a multicellular “body.”
As for organs and organ systems, I offer the siphonophores. They are colonial cnidarians (basically jellyfish) that form colonies of individual cloned animals. What’s interesting is that certain animals in the colony specialize for certain tasks. There are polyps that specialize in digestion, medusae that specialize in “swimming” and pulling the whole thing along; in Portugeuse men-of-war there’s a polyp that acts as a “float” for the whole colony.
Siphonophores are probably my favorite animals, precisely because they so weirdly exist on this boundary between individual and group. I think if you study them and think about the implications, the idea of something evolving dedicated organs and systems won’t seem all that strange. They are also amazingly and weirdly beautiful.
To answer your question more abstractly—first of all, you have to realize that complex, multicellular creatures with organs and systems are extremely rare in the grand scheme of things. Almost every single organism that exists is a bacteria. Even cells with nuclei are comparatively rare.
Why does complex stuff exist in the first place? Because it works. Maybe another way of asking this question is “how do cooperative systems evolve”? Because multicellular and complex organisms can really be thought of as individual components somehow working together as a coherent whole. But nature is filled with such examples, and when they happen they are incredibly successful. Social insects (ants, bees, termites) are by far the most successful insect species. Cooperation is hard and rare, but evolution seems to favor it when it happens.