Like an old piano can go out of tune in the cold weather, so our skin can need extra attention too. In cold weather the air around us, outside and indoors, holds less moisture: the relative humidity goes down. Then water contained within a semi-permeable membrane like skin will tend to escape fast into the surrounding air. See How emollients work. The old piano has wooden pegs inside that hold the strings tight in the sound board in warm, moist weather. But they dry out and become loose in cold, dry weather. Our skin is the same. The skin dries out, becomes itchy, and acute eczema is more likely. So, more frequent moisturizer applications per day are needed in cold weather to prevent the skin from drying out.
Central heating also dries the air. A humidifier may be useful as a remedy - it will also reduce the frequency of the piano tuner's visits! However, high humidity and limited ventilation is associated with increase in house dust mite activity - what this does to pianos is not reported, but it is associated with increased severity of atopic eczema.
It seems better, and cheaper, for the skin to use moisturizers more often in cold