It's been a very long and gradual transition. It's important to note it's not clear how common polygamy ever really was. Biblical characters are depicted having multiple (or even very large numbers) or wives. Even if they're not historically accurate one could think that they depict historical reality for the writers, but they could be only accurate for a small segment of the population (in the biblical writers' society, powerful men had lots of wives) or representative of views of the past (biblical authors thought that people in the past had lots of wives).

Anyway, during late antiquity (roughly when the Talmud was compiled), monogamy seems to have been the norm, and probably preferred, but polygamy was definitely a theoretical possibility. Later on, in the Middle Ages, polygamy was frowned upon and discouraged by Rabbis in their writings, if they allowed it at all (more on that later). However, in Muslim-majority countries Jews did occasionally still practice polygamy (mostly in Yemen, where there was a large Jewish community and the polygamy existed in the dominant non-Jewish culture) until quite recently (the 20th century), when those communities migrated en masse to Israel, where the practice is not allowed (though multiple marriages performed elsewhere are registered legally with no legal penalty). Though rare, it ended completely within recent memory.

However, for many Jews polygamy has been outright forbidden for several centuries. In the Middle Ages Rabbeinu Gershom of Mainz made a series of rules formalizing existing preferences and customs as law, which were agreed on by consensus. Because of his prominent role, and Rabbis over time agreeing with him and formalizing his pronouncements, they became standard requirements for all European Jewry (which is the dominant group in the US and the majority or Jews in Israel).

His most important two rules (for this topic, anyway. Another was the prohibition on reading others' mail) were the prohibition of intermarriage and requiring a woman's consent in divorce. Both of these were definite preferences earlier, but beginning with him they were absolute rules, which spread through European Judaism.

So it's a bit of a mixture. In European Jewry there was a conscious decision to make it a rule, which was preceeded by social convention and gradually spread. In Middle Eastern Jewry, it was more of a social convention, and the practice ended quite recently with it being against the law, but there still is no formal religious prohibition on it for them.