The easiest - and most likely answer - is if they are displaced by another group (likely by warfare). Or, similarly, if they need to travel but don't want to go through another group. Another option is for trade or easier transport.
The habitable zone will be a thin ring, so the only way any civilization can expand is by moving north or south. Civilizations will naturally seek to expand - there's not many civilizations that are perfectly stable over the long-term. Surplus food allows surplus population, and that allows expansion (especially if the surplus of food disappears, so they need to spread out to get more farming land). So, it's safe to assume that there will be some form of competition or war.
If a group is pushed out of its land, it will likely flee east or west into the uninhabitable zones.
For the trade/transport scenario, we shouldn't assume that the easiest travel is along the habitable zone. Maybe there are mountain ranges or rough seas that cross that zone, so the easiest way to travel is to venture out of that area and into the less-hospitable places. And maybe the planet has oceans or rivers - so liquid water would be present in areas that may otherwise be uninhabitable (and the water will moderate the temperature of the land around it). If the geography is right, in some places there could be a continuous flow of water into the hot side and a continuous flow of storm clouds back to the cold side. These waters would be very difficult to navigate (they would probably be very fast), and the storms could be very dangerous. So it would be safer to route around this area, venturing far into the desert side (which will actually be something of an oasis because of the constant water) or into the cold side (which may be more like the UK due to the constant warm air and frequent rain) to avoid those. A river that crosses perpendicularly from cold side to hot side would create these climate patterns - and the hot side will always have a lower water level because its water is evaporating, so the river will always from from cold to hot.