Why is a Muslim husband required to have a bath after having sex with his wife? Why is merely washing the relevant areas not enough? Is such an intimate relationship a dirty thing to do?
The real message of Islam is purification. Islam wants a person to remain physically as well as spiritually clean. However, the extent of cleanliness required when we become physically impure in certain situations cannot be discovered or ascertained by human intellect. In such situations, it is the Sharī‘ah of the Almighty which guides us and prescribes a way. For example, it tells us that the extent of cleanliness required to become ceremonially pure after urinating or passing a wind is to do Wudū. Similarly, it tells us how to become ceremonially pure after having an intimate relationship with one’s wife. Whether the husband and wife after having such a relationship should just wash the relevant parts, do Wudū or have the ceremonial bath are matters in which we ourselves cannot reach a decisive conclusion.
This is similar to the fact that we have been asked to pray five times or asked to fast for a month. In these cases also, we cannot ascertain ourselves the most appropriate number. Thus if the Almighty has said that the basic objective of the prayer is to remind a person of keeping Allah in his remembrance (See 20:14), it is only He the Creator of human nature who knows that five prayers (and not any another number) is the required number to produce the optimum result. Likewise, it is only the Almighty who can tell us that fasting a month (and not for some other period) is the required period to nurture piety in a person which is the basic objective of fasting (See 2:183).
As far as the last part of your question is concerned, it is to be noted that an expression of the animal instincts found in a person makes him unclean to a certain extent. He washes or cleans his hands after having food. Similarly, ejaculation calls for cleanliness in a prescribed way. However, this does not at all mean that eating or drinking or having an intimate relationship with one’s wife are intrinsically dirty acts. They are the needs of a human body. However, they make a person physically unclean and therefore, require purification.