Learn Spanish (Latin America): La gente se puso a abuchear al payaso.

You absolutely could use empezó a. However, ponerse a followed by an infinitive is a very common phrase that means someone got to work on something or started doing an action, often with a sense of suddenness or effort. Because the crowd suddenly started booing, se puso a abuchear is a perfect fit.

Why is the verb puso singular when "people" implies a plural group?

In Spanish, la gente is a singular collective noun, much like "the crowd" or "the family" in English. Even though it refers to multiple individuals, grammatically it counts as one single entity. Therefore, you must always use the third-person singular form of the verb with it. That is why we use puso and not pusieron.

Doesn't the verb ponerse usually mean "to put on", like clothing?

Yes, ponerse is frequently used to talk about putting on clothes. It can also mean "to become" (like ponerse triste for "to get sad"). But when you use the specific formula ponerse a + an infinitive verb, the meaning changes completely to "to start to do" or "to set about doing" something.

Why is it al payaso instead of just el payaso?