Learn Italian: Gli struzzi ci hanno inseguito per beccare il pane. - The ostriches chased us to peck the bread.

With the third-person direct object pronouns (lo, la, li, le), past participle agreement is mandatory. However, with the first and second person pronouns (mi, ti, ci, vi), making the past participle agree is optional. Both ci hanno inseguito and ci hanno inseguiti are perfectly correct in modern Italian.

How would we use this lesson's grammar (mettersi a) to say the ostriches started chasing us?

You would say: Gli struzzi si sono messi a inseguirci... (The ostriches started to chase us...). Since mettersi is a reflexive verb, it requires essere as the helping verb in the passato prossimo, and the past participle must agree with the masculine plural subject, becoming messi.

Why do we use per before beccare instead of a?

When you want to say "in order to" do something, you generally use per + infinitive to express the purpose of an action. While a + infinitive is used to express purpose after a few specific verbs of movement (like andare or venire), the verb inseguire (to chase) doesn't trigger that exception, so we use the standard per.