Learn Italian: Il branco si è messo a inseguire la macchina.

Because the verb used in this construction is the reflexive mettersi (to start / to set about doing something). All reflexive verbs take essere in compound tenses like the passato prossimo, and the past participle must agree in gender and number with the subject (il branco is masculine singular, hence messo).

What is the difference between using mettersi a here versus ha cominciato a or ha iniziato a?

While all of them translate to "started to", the expression mettersi a emphasizes suddenness, a spontaneous change in behavior, or setting oneself to a task. In this sentence, it paints a picture of the herd abruptly and actively deciding to chase the car, rather than just neutrally beginning an action.

Since il branco implies many animals, why isn't the verb plural (si sono messi)?

In Italian, collective nouns like il branco (the herd/pack), la gente (people), or la famiglia (the family) are grammatically singular. You must always use a singular verb to agree with the grammatical number of the noun, never the logical number of individuals inside the group.

Can branco be used for groups other than a "herd"?