Imam Hussain (AS) wanted sincere eye witnesses to tell people after his martyrdom what really happened in Karbala keeping in mind the falsehood which Yazeed and his gang were intending to spread among Muslims against Imam Hussain (AS). Mo'awiyah, father of Yazeed, spread rumors among people in Syria that Ali does not pray Namaz, that is why when the news reached Shaam that Imam Ali (AS) was martyred inside the Masjid of Kufa, people of Shaam started asking: What Ali was doing in the Masjid when he does not pray Namaz?
Yazeed and his gang wanted to fabricate stories against Imam Hussain (AS) to mislead people away from Ahlul Bayt (AS). Imam Hussain (AS) took his family members with him knowing what will happen to him and them, to make them witness and convey the message to all Muslims after him.
The great rule of Lady Zainab and Lady Um Kolthoum after Karbala shows how it was necessary to have eye witnesses to inform Muslim generations and refute the false propaganda of Yazeed.
The sufferings of the family of Imam Hussain (AS) was mentioned by him as narrated when he replied those who asked him why to take his family members: Allah has ordained to see me martyred and see my family members captured شاء الله أن يراني قتيلا ويراهنّ سبايا