Accompanying women and children in Karbala:

1. So that the uprising and movement of the Imam and other martyrs of Karbala will not be fruitless and the message of Ashura will reach other Muslims and the people of the world. If the Imam did not take his family to Karbala, the Yazid regime would have removed the Imam from his path and would have put the Imam's blood on the necks of his opponents and would have destroyed them as well under the pretext of avenging the Imam's blood. Even now, Yazid's followers have spread rumors among Muslims that Imam Hussain (AS) was invited to Kufa by his Shiites.  They surrounded the Imam and martyred him and his companions.  now they are lamenting the sin of killing Imam Hussain (AS) and  beat themselves.

2. If Imam Hussain (a.s.) did not take his family to Karbala, Yazid and his governors in Mecca and Medina would take the Imam's family hostage to pressure the Imam and force him to pledge allegiance.

Imam took his family with him to Karbala. The Imam's (peace be upon him) family also brought the Imam's uprising to fruition. The sanctity of the caliphate system was broken and Muslims realized that the caliphs were not equal to the Prophet (PBUH) and were not on his path. Rather, they are usurpers and heretics who are gradually destroying Islam.

 As a result, uprisings against the Umayyads started from all sides, and finally, the continuation of these uprisings led to the destruction of the Umayyad dynasty. [1]

[1] Seyyed Ibn Tavus, Lahof, Al-Alam, p. 84.