Why did eukaryotic cells develop?
As far as I understand your question, the "development" of eukaryotic cells can be linked to the endosymbiotic theory Symbiogenesis. 

This diagram only depicts a possible trajectory of evolution which is still unknown.




A cell that respired anaerobically took in a bacterium that respired aerobically, supplying both itself and the larger cell with energy in the form ATP. This created a competitive advantage for the larger cell because aerobic respiration is known to be more efficient than anaerobic respiration. Unhurriedly, the aerobic bacteria evolved into mitochondria and the larger cell into a heterotrophic eukaryote. 
A heterotrophic cell took in a smaller photosynthetic bacterium, which supplied it with organic compounds, thus making it an autotroph. The photosynthetic prokaryote evolved into chloroplasts and the larger cell into photosynthetic eukaryotes.