If two objects with different temperatures are in contact, heat is transfered naturally from the one with higher temperature to the one with lower temperature. This makes th object with higher termperature to coll the the object with lower temperature to heat.
In this case, we have ice water in contact with a spoon in room temperature. Since the spoon get colder, it means that heat was transfered from the spoon to the ice water, decreasing the temperature of the spoon and increasing the temperature of the ice water. For this to happen, the room temperature has to be higher than the temperature of the ice water.
So, what explains what happaned in that the room temperature is higher than the ice water temperature, so, the spoon transfer heat to the ice water, getting colder.