After the followers of the Prophet were forced to leave the Byzantine Empire they first resettled in Syria, taking the Divine Source with them. However, Trinity followed them there, tried to murder the Prophet, and drove his followers out. The Exodus continued and they turned north.
After a long and sorrowful journey they finally found a place to settle. The Prophet and the five founders (Kokkos, the Astronomer, Damien, the Prophets's Forge, Aurora, the Physician, Rivkah, the Architect, and Valentinus, the Time Keeper) build the City of Kitezh in a remote valley deep in Siberia. They hoped Trinity would not follow, especially since they were supposed to believe that the Prophet was dead and buried in Syria.
The city flourished for a century or longer until the Mongolian hordes - accompanied by a Trinity agent - threatened its borders. The inhabitants of Kitezh barricaded themselves in the City. When the Mongols attacked and it was obvious that the city was about to fall, Jacob used the Divine Source on the City Guard, the Athanatoi, to turn them into the Deathless. They defeated the Mongols, but at a terrible price: They brought the glaciers down onto the city, killing invaders as well as inhabitants.
The City was lost and the Remnant inhabitants were forced to survive in the harsh environments of the Siberian wilderness. Only a few of them survived in the Geothermal Valley, where they lived off the land by hunting and agriculture.
Kitezh has survived to current days only as a myth. Lara's father searched for the Divine Source and the Prophet for a long time. After his death and her recent supernatural experiences on Yamatai Lara continues his research. She travels to Syria and discovers the Tomb of the Prophet to be empty. She concludes from her father's research that the Lost City of Kitezh lies in Siberia and travels there, yet Trinity arrives before her.
After meeting and befriending the Remnant, Lara succeeds in entering the City after finding the Atlas, an ancient map, with the help of Jacob. Still preserved under the ice, but guarded by the Deathless, it gets heavily damaged during the fighting, but Lara manages to prevail and in the end destroys the Source.
It is not clear what becomes of the City after Lara leaves, whether the Remnant - after Jacob's death and under the new leadership of Sophia - reclaim it or if they leave it behind, no longer needing to protect it and the Divine Source. There is talk among the Remnant, discussing their options, whether they should leave and live in the modern world, or if they should stay and continue their traditional life. Lara is able to recover most of the Ancient Knowledge from the ruins and teach it to the Remnant.
Real World Version
The in-game city is based on the real-world myth. The City of Kitezh is supposed to lie under the waters of Lake Svetloyar.