Since, woman has no financial obligation and many economical responsibilities lay on the shoulders of the man; the rationale behind the Islamic ruling on inheritance is to implement a sort of balance between the rights and responsibilities of each group.
To answer the question that why Muslim woman's inheritance is half of man we must, consider the responsibilities Islam has placed on the shoulder of men. It necessitates his spending money, much more as compared to the women, in order to carry his responsibilities out. Hence, man is in need of wealth and money even many times more than a woman while a woman’s inheritance is only half that of a man’s.
To briefly mention some of the reasons, why the inheritance of man is more than that of woman's, are as follows:
First, the upkeep of women is upon the shoulders of men, according to which a man is responsible to look after all the households' pays including himself, his wives, children and sometimes the expenses of his parents, brothers and sisters etc. But a woman does not have such responsibilities and she only takes her portions from the inherence as well as the dowry from her husband for her own savings. In fact, the portion of the inheritance of the woman is for her own savings, but the portion of man is not the same; rather; his portion must be spent on the expenses of his daily life and that of his wives and children.
Second, the above mentioned financial responsibilities are not the only ones that must be fulfilled by man; rather, he has many more other responsibilities. For instance, he must spend money in the way of war, or if one of his relatives accidentally kills someone or injures someone, he must pay certain monies to the family of the victim, whereas a woman has no responsibility in this regard.
Third, before a woman is married it is the duty of the father or brother to look after the lodging, boarding, clothing and other financial requirements of the woman. After she is married it is the duty of the husband or the son. Thus, Islam holds the man financially responsible for fulfilling the needs of his family.
In conclusion, In Islam a woman has no financial obligation and the economical responsibility lies on the shoulders of the man. It seems that the share of a man in inheritance is two times that of a woman, but in all practicality his share in the sum total of personal wealth is far less than hers. That is to say that the main rationale behind this Islamic rule is to implement a sort of balance between the rights and responsibilities of each group.