Why has Islam permitted man to beat the woman?

Muhammad (may Allah's peace and blessings be upon him) has never beat a woman in his whole life. As for the Qur’anic verse that talks about beating, it refers to non-violent beating in case of recalcitrance. In some period, this kind of beating was depicted in the man-made law of the United States of America as the permissible beating that should not leave trace on the body. That is used to prevent a graver danger like when someone shakes his son by the shoulder to wake him up from his deep slumber so as not to miss his exam.

Let us imagine that someone found his daughter standing on the edge of the window to throw herself; he will inadvertently grab her and pull her back so as not to hurt herself. This is exactly what is meant here by beating the woman; it is the husband's endeavor to prevent his wife from destroying her home and her children's future.

This, actually, comes after several stages as mentioned in the verse that reads:

{As for those women on whose part you fear rebellion, [first] admonish them, [next] forsake them in bed, and [last] hit them [gently]. Then if they obey you, do not take further action against them. Indeed, Allah is Most High, All-Great.} [211] Surat an-Nisā’: 34.

Considering the woman's weakness in general, Islam has granted her the right to resort to the court in case of being mistreated by her husband.

The marital relationship in Islam is to be built, in principle, on love, tranquility, and mercy.

{And among His signs is that He created for you spouses from among yourselves that you may find comfort in them, and He placed between you love and mercy. Indeed, there are signs in this for people who reflect.} [212] Surat ar-Rūm: 21.