Prior to the Easter/Western doctrine schism, married priests were not uncommon; this followed in the traditions of the apostles being married and as the cardinals and, to a lesser extent, the priests were considered to be spiritual successors to the apostles. There were restrictions on them (and in the Eastern and Oriental traditions today, married men can become priests, but unwed priests cannot marry).
Then the Middle Ages happened, and vows of celibacy became the norm. Part of this was the control the papacy wished to exert over its clergy (a sort of, "If you're married, we don't have your full devotion, because your work will come second" line of thinking, designed to make them even more subject to the Holy See), and part of it was a restructuring of paradigm:
- The Virgin Mary, a core tenet of Catholic faith, needed to be represented by a virgin priesthood
- The Bible, while not outright banning marriage, advocated celibacy as the preferred state
- They didn't want priesthood becoming a hereditary position, as it was for Jews and pagans
- Virginity meant purity of body and spirit
There's more, but the gist of it is a combination between controlling the mouthpieces through which the papacy spoke to the people by restricting their outlets for devotion, as well as a sort of face lift for the clergy.
Further reading found here
I completely agree that the priests should be allowed to marry. But married monks or nuns? It's impossible.
By strict definition, Catholic priests are