Why is a woman not allowed to pray while in the state of Haydh? Despite the state of impurity why would Allah prevent her from worshipping Him?
It is Almighty God who appoints obligatory acts to His creation, and for us Muslims to comply to the commands and requirements given to us by Him. If the Almighty says that we pray in a particular way, and in another way when travelling, or to fast, and at times not fast, we must follow this.
Islam is a natural, understanding and forebearing, and the Shari'ah accommodates to our human needs.
The condition of prayer is that one must be in the state of ritual purity, which means one cannot pray while in the state of Janabah, Haydh or Nifas.
As for the exemption given to a woman being in Haydh or Nifas (post-natal bleeding), one reason that can be given is that a woman goes through various physical, emotional and mental strenuousness while menstruating. Almighty God wants to give her a rest from fulfilling this particular obligation.
It could be difficult for her in these states to bow and prostrate, and so on, so she can rest from these actions for the few days. She is bleeding, and a lot of things are being affected by this, so she should not add burden to this by having to pray, or having to fast.
However, in no way does it mean a woman is to be "isolated", or that she is "dirty", or that she will be deprived of socialising, or worshiping Allah ta'ala during this time. These are things that other religious have mentioned regarding a menstruating woman.
Rather the opposite, it is still recommended that at the five wajib prayer times a menstruating woman performs her wudhu', and sits in her prayer niche and supplicates and does dhikr, to the amount of how long the obligatory prayer would take. Also, she can recite the Quran (with exemption of certain surahs), and anything else that will grant her spiritual proximity to the Almighty.
And Allah knows best.