There are many theories as why eukaryotic cells have membrane bound organelles, but the most common one is the endosymbiotic theory.
This theory suggests that eukaryotic cells came about via endocytosis, which is when one cell engulfs another.
Assuming eukaryotic cells come from prokaryotes, the theory says that one prokaryote engulfed another, and instead of being digested, the engulfed cell remained intact and functional. The cells then formed a symbiotic (mutually beneficial) relationship. The cell which was engulfed then became the first organelle.
Beyond the origin of organelles, they serve the purpose of more specific cell functions. Eukaryotic cells are far more specialized and advanced than prokaryotes, and organelles help with intercellular (between cells) communication, which allows multicellular organisms to develop.