An object gains energy when it absorbs radiation and loses energy when it emits radiation. Both happen all the time. Power is the energy transferred each second, measured in watts (W), so we can compare the power absorbed with the power radiated.
For an object to stay at a constant temperature, the average power it radiates must equal the average power it absorbs. The energy leaving each second is then replaced by the energy arriving each second, so its internal energy, and therefore its temperature, does not change.
The word average matters. The radiation arriving may vary, for example as clouds pass over, so the temperature can wobble slightly but stays steady overall if the averages are equal. A balanced object does not stop emitting. Both emission and absorption carry on, and they simply cancel out in their effect on the temperature. Thinking only about radiation, a mug of tea left in a room ends up at room temperature because it eventually absorbs as much from the room as it radiates.