Why does Torah law allow polygamy?

Just to magnify your question somewhat, you'll note that Torah presents the original paradigm of marriage – that of Adam and Eve – as monogamous. Furthermore, virtually every instance of polygamy recounted in the Torah is related directly by the narrative to some sort of calamity—whether strife between competing wives, as was the case with Hannah and Peninah,1 or between rivaling half-siblings, e.g. Jacob's2 and King David's sons.3 Even the very verse4 in which the Torah provides a green light for polygamy frames it within an undesirable circumstance: "If a man will have two wives, one beloved and the other hated..."

Why then make room for trouble? If the ideal union of man and woman is an exclusive one, why should a "nation of priests and a holy people" compromise?

The simple answer is that Torah deals with life on earth, and the gamut of social life and human experience over all of history and world geography is too diverse to be restricted to one narrow ideal. Take, for example, an agrarian society whose male population has been decimated by war. How are women to survive and how is the population to replenish itself without the mechanism of polygamy? Similarly, a man married to a barren woman who could not produce sons to help in the field and defend the fort would find himself ill put to survive in those times. In an exclusively monogamous society, his wife would find her position insecure. Although, in normative circumstances, being "only one of many" compromises a woman's value as a person, in these situations a permit for polygamy is a form of compassion.

The only case of a polygamous rabbi recorded in the Talmud5 provides an excellent illustration: Rabbi Tarfon married 300 women. Why? Because there was a famine in the land. But Rabbi Tarfon had plenty of food, since he was a Kohen and received the priestly tithes. The wife of a Kohen is also permitted to eat those tithes. Those 300 women were very happy that the Torah permitted polygamy.

Torah discourages abuse of this permit—not just by recounting the calamitous narratives mentioned above, but also by placing requirements on the husband. For every extra wife, no matter how lowly her status, a man must provide "food, clothing and conjugal rights" commensurate to her needs, his capacity and equal to any other wives.6 Additionally, the husband must provide separate housing for each wife. Divorce requires involvement of the scribes, and the sages later instituted the ketubah as a further impediment of divorce. (See also Why is Jewish Marriage so One-Sided?) We see that these means were in fact effective, polygamy in Jewish circles was historically a rare exception.

Rare, but necessary nevertheless. Even when Rabbi Gershom and his Rabbinical Court assembled to create a ban on polygamy due to the conditions of their time (see previous link for more on this injunction), they nevertheless left the door open for extenuating circumstances. That loophole has proven vital in many an instance—for example, the case of a wife who has become (G‑d forbid) mentally incapacitated and is not halachically qualified to receive a divorce.

You may wish to think of Torah as the DNA of a highly resilient organism called the Jewish People. Whenever circumstances change, this organism looks back into its DNA and finds some code that allows for an adaptive modality. There's plenty off limits, but there is enough leeway to provide for every situation human life on planet earth can throw at you. Proof is, we've been through it all – nomadic, agrarian, civilized, industrial, technological – and in every part of the world, and we're still here, strong as ever.