In short
An adiabatic process is a change in which no heat enters or leaves the system, so . The system changes its temperature only because work is done on it or by it.
From the first law, . When a gas expands adiabatically it does work at the cost of its own internal energy, so it cools. When it is compressed adiabatically, work is done on it and it warms. A bicycle pump gets hot for this reason. For an ideal gas, , where .
A gas expands adiabatically and does of work. No heat flows, so its internal energy falls by and its temperature drops.