What is the main purpose of Horatio's speech?

to explain to the audience how the former king died and became a ghost

This is to persuade the gathering of people of the truth of the ghost that has showed up before Marcellus and Barnardo on two earlier evenings. He is the suspicious logician who does not trust in ghostly visions and unquestionably not in phantoms, and when he is persuaded that a heavenly being has appeared to him and his associates, the gathering of people is persuaded also.