I don't know if we have all the answers to this question, but I can offer some hypotheses. Basically, eukaryotes had adaptations better suited for large multicellular organisms.
The first of these would be based on their internal organelles, such as mitochondria, nucleus, plastids, vacuoles, etc, and the internal compartmentation they provided. The mitochondria provided an energy provucer superior to any bacteria had, and chloroplasts provided a superior means of producing food. I suspect that the compartmentation provided by organelles also contributed by allowing for large differences in ion concentrations between areas of the cell. Calcium is an excellent example of this. While compartments such as vacuoles and endoplasmic reticulum are relatively high in calcium concentration, the cytoplasm is normally held to very low levels. Thus, calcium can be used as a signal transducer when allowed into the cytoplasm, where it activates various proteins resulting in a cascade effect.
Another big difference between prokaryotes and eukaryotes is that in eukaryotes the DNA is stored in a unique configuration with proteins (such as histones) in what are called, chromosomes. This allows for much finer regulation and control over transcription of the DNA, than exists with the naked DNA found in bacteria.
Thus, eukaryotes were better equipted to support the energy requirements and fine control necessary to coordinate and regulate the activities of mulitcellular organisms with many different cell types, tissues, organs, and eventually, organ systems, as we humans have.