Why them, not the other two tribes?
The Kingdom of Israel split into two after the reign of Solomon, the Northern Kingdom of Israel and the Southern Kingdom of Judah, which survived into historical mentio (Judah is mentioned in an Assyrian tablet from the 730s BC, and Israel in the Black Obelisk of Shalmaneser III a century earlier). Territory in biblical Israel was apportioned by tribe, and the tribe of Judah had the largest swath of land in the South, which explains the territorial divisions of the split Kingdoms.
The Assyrians conquered Israel but did not manage to conquer Judah in part due to the alliance formed with Egypt, and eventually Judah became a tributary state of the Assyrian empire and avoided the same destruction and scattering that Israel experienced. Eventually the Babylonians, who had supplanted the Assyrians as the dominant power in the region, conquered Judah and exiled the inhabitants to Babylonia (hence the term Babylonian exile), where they remained united as a community. Seventy years later, the Persians supplanted the Babylonians and let the Jews return to Israel/Judah.