Mythology:
There is this story of Devendra acquiring brahma-hatya dosha due to killing Vritrasura, as a result of which he lost the position “Indra”. The brahmahatya was later shared by four:
women (their menstruation is when they exhaust this share of dosha),flora (the white gooey thing - that is why such flowers are not offered for worship),water-bodies (I have heard it is the foam - they say touching such foamy waters is not allowed, with one or two exceptions)Bhu-devi (she exhausted it when Krishna was born).
Because a woman is exhausting a part of what is brahma-hatya (and she herself is subject to it) during those days, she is not fit to enter temples.
Rational thought:
Women are not disallowed from just entering temples, but just about any free movement is restricted. In fact, menstruating women are vaulted during their periods. I have heard the following reasons from various sources (pravachanams, grandmas, priests, other elders, etc). I haven’t come across texts, but such is also the nature of dharmasastras. Perhaps other upavedas & sastras might have some sources.
Sanitation: Our ancestors have been paranoid about hygiene, public health and sanitation. This is just one more, especially during days when sanitary napkins and tampons were unavailable. They were also keep away from other public areas to minimise any spread of germs.Women’s health: There is a school of thought that women need rest during those days and doing hard work was disallowed to ensure their fertility. This makes sense especially back when women walked miles on stone grounds to fetch water, firewood, etc and were exposed to sun and rain.Tantra: This one is a very close cousin to (2). There is a belief that vibrations from temples, vedic chanting, etc can affect women’s anatomy in some ways. In fact, there are special chanting for pregnant women, etc too. Also has something to do with kundalini, yogasastras, etc. That is why some asanas are disallowed for women during their periods, pregnant women, etc.Moods: There is a school of thought that one’s moods affect those physically around them as well. And women’s moods are quite weird during these days. Therefore, they have been kept away from worship in order to ensure the harmonious atmosphere, ambience, vibrations (whatever you want to call it) in the temples and places of worship.Abstinence from sex: This is two fold. One could be that they thought it was not in the woman’s welfare. The other could be related to (4), when rajas is a little more prominent in women during those days. garments to wear that looks sexy in the cocktail
In fact, all related rules like restrictions on food (pickles and preserved food are disallowed), not allowed to take bath in water-bodies, not allowed to wear collyrium on eyes - all mostly tie back to some form of women’s health and public hygiene, as was understood back then.
Now all this said, women are NOT disallowed from worshipping on those days. There is no restriction to sing the praises of Lord. Nothing can stop you from offering worship in your mind. So, it is not really impure in a sense. It is only entering common spaces and chanting of some mantras that seems to be disallowed. Chanting nama, singing bhajans in one’s own place is not disallowed at all.