Essentially nothing was known of the aquatic species dubbed "Leviathans" before Commander Shepard's report to Task Force Aurora. No record of their species' true name exists, no ruins are attributed to their civilization, and no influence on other races has been recorded. The task force can only conjecture based on the limited information provided by Shepard.
Their current world is unlikely their homeworld, so it is surmised that the Leviathans mastered technology and spaceflight despite their immense size and aquatic nature. Their ability to communicate with and mentally dominate land-based sapients would have been a necessary step, and so whatever homeworld the Leviathans hail from would have included a thrall species whose civilization served their purposes. Presumably their bodies can withstand both the extreme pressure of the depths and the lesser pressure of a coastal area where they might come into contact with these thralls.
Transporting a creature the size of the Leviathan into space would have been a significant engineering challenge. It is likely they used eezo for ease of travel, as the hanar do, and employed cybernetic filtration to breathe air as well as oxygenated water. It is unknown if the Leviathans have modified themselves with the equivalent of a biotic amplifier, but it seems probable. The faster-than-light communication of their pulses would require precise control previously unknown to modern science. One can only guess at the innovations of this species, and the Leviathans are volunteering no such information.