Maybe it doesn't? Perhaps you see more steam (ie. condensed water vapour) because the air is colder and the water vapour from the bath consenses in the cold air.
But maybe it does! When the air is colder, the amount of water vapour it can have is less. (Which is why it condenses, as above.) But if the air gets warmed by the bath water then it becomes capable of holding more water vapour. So maybe more bathwater can evaporate then?
Since water vapour is invisible and the amount of water which evaporates is too small to notice as a fall in liquid level, my guess is the first case. You are noticing more condensed water vapour (colloquially, "steam") in the cold air and thinking that is water vapour. When the air is warm, the water evaporates, becomes invisible and you never see it again! So you don't realize there was lots of water vapour.