As the air cools it is capable of holding less moisture and with the same amount of moisture present the RH will increase. E.g. if the air is capable of holding 15gm of water vapour per kg of air and has 5gm/kg its RH is 33.3% but if the air is cooled it will reach a point where it can only hold 10gm/kg and it still has the 5gm/kg present its RH will have risen to 50%.

Warm air and cold air can be dry or moist; it is the change in temperature which results in a change in RH.