Answer : This is because heat flows more easily through metals.
Suggestion : The riddle is more illuminating if the question is rephrased as follows:
"When a steel cot (good thermal conductor) and a wooden cot (poor metal conductor) are both at the same temperature, why does the steel cot feel colder?"
Explanation : It is a common misconception that our perceptions of warmth and cold are directly related to temperature or temperature-difference between our body and the object. In reality, it is related to heat. Heat is the flow of thermal energy . When heat flows into our body we perceive it as warmth and when heat flows out of our body we feel it as cold. Higher the rate of heat flow warmer/colder does it feel. Metals being good thermal conductors allows heat to flow easily and so feel colder.
Thermal Energy*, Temperature and Heat** are related concepts and one is often misunderstood for another. A clear understanding of these concepts will help one to understand this question.
Thermal Energy : According to the kinetic theory of matter, matter is made of a very large number of particles called atoms/molecules. These atoms/molecules move in space, rotate and vibrate randomly. There is kinetic energy involved with such random motions. The sum total of kinetic energies, associated with random motions, of all the atoms/molecules that make up the system is called Thermal Energy .
Temperature : Since atoms/molecules collide with each other, there is always transfer of kinetic energy from one to another. Some are moving fast while others are moving slow etc. The speed, and hence kinetic energy, of any atom/molecule is a random variable which changes unpredictable with time. However one can define an average value for the speed. Temperature is related to the average kinetic energy associated with such random motions. It can be seen as a concentration of thermal energy. Where the thermal energy is more concentrated, the temperature is high and vice versa.
Heat : When the concentration of thermal energy differs from one point to another, there is a flow of thermal energy from regions where they are more concentrated (higher temperature) to regions where they are less concentrated (low temperature). Heat is the flow of thermal energy. The flow of thermal energy stops when the concentration of thermal energy (Temperature) is homogenized. When a system attains this state it is said to heave attained Thermal Equilibrium. As long as there is temperature difference there will be flow of thermal energy. Our perceptions of warmth and cold is related to heat.