The use of "they" as a singular pronoun has recently been part of a culture war, but as the linguists quoted here point out, we've been doing it for hundreds of years. Beginning in the 1500s, grammarians began to lobby for using "he" as a universal generic pronoun, but usage of "they" still persisted — not just in everyday language but in the prose of novelists like Jane Austen. "They" has survived — and, now, resurged — because it's a pretty useful linguistic strategy, basically.
The piece is great (go read it all!) and here's a snippet:
The word has a long, complex history, and today it exists in several nuanced forms. Evan Bradley, a linguist at Penn State Brandywine, says that "not all singular 'theys' are the same." And they enjoy varying levels of acceptance. Some are so common that most people find them unremarkable. Imagine that a random driver swerves in front of you while cruising down the road. You might respond, "They cut me off!" in reference to the singular driver. Or, perhaps you find a billfold with no identification inside? It's natural to say, "Somebody lost their wallet."
The oldest kind of singular "they" appeared in writing around 1370, and certainly showed up earlier than that in speech. Shakespeare used it frequently around the turn of the 17th century ("And every one to rest themselves betake"), and Jane Austen in the 19th ("No one can ever be in love more than once in their life"). In these examples, the antecedent — the noun to which the pronoun refers — is generic, meaning it doesn't indicate a specific person, or gender for that matter.
Despite these precedents, grammarians steered the generic antecedent in a different (and more masculine) direction. In the mid-1500s, William Lily declared in his Latin textbook that "the masculine gender is more worthy than the feminine, and the feminine more worthy than the neuter." (He was speaking in the context of grammatical gender, rather than biological sex or socially-constructed gender roles in humans. And notably, English as a language is mostly without grammatical gender). Lily's book stood as the foremost Latin grammar guide for centuries, and his gender rule, transferred to English, lived on in the axiom that "the masculine embraces the feminine."
That thinking gave rise to the generic "he," a longstanding linguistic scourge upon feminism. These days, most writers who can't stand singular "they" at least sprinkle their writing with generic "she" as well as "he." But for a long time, the masculine version was ubiquitous, pervading all writing up to and including the Constitution of the United States of America. Article II, Section 1, in laying out the president's powers, states that "He shall hold his office during the term of four years."