Why didn't Hussain bin Ali leave Karbala'?

Why did Hussain bin Ali go to Karbala' knowing what was going to happen to him & his family?

First Answer :  
Yazid (la) had sent spies to murder the Imam (as) in the House of Allah and al-Husayn (as) had to change his niyyah from Hajj to `Umrah to prevent the violation of the sanctity of the Haram. References for this can be found in the books I am listing for the answer to your question below.
al-Hurr b. Yazid al-Riyahi, under instructions from Ibn Ziyad (la), was responsible for leading the Imam to Karbala', even though the Imam had been headed to Kufah. All books of Islamic history that I am aware of more or less convey the following account:
"Al-Hurr said, I desire to take you to the commander Ubaydullah. Imam said,
By Allah! I shall not do so.
Al-Hurr said, By Allah! I too shall not leave you. They repeated this thrice, and when their conversation got heated, Al-Hurr said, I have not been commanded to fight with you. I have been ordered to remain with you until I take you to Kufa. Then now when you refuse to go to Kufa, take a way, which does not go to Kufa nor Madina, while this being a compromise between us. Then I shall write a letter to the commander and you write to Yazeed or Ubaydullah, and Allah should bring forth fairness, so that I may not be entangled in your matter. Imam turned his horse towards Qadisiyyah and Uzayb towards the left, and Al-Hurr and his followers travelled by their side."
http://www.maaref-foundation.com/english/library/pro_ahl/imam03_husain/nafasul_mahmoom/20.htm (p.180)
Once they had arrived in Karbala', a letter written to him by Ibn Ziyad (la) stated:
""Now then O Husain! I have been informed that you have haulted at Karbala. Yazeed has written to me not to recline my head on the bed and not be satiated until I send you to Allah, or you submit yourself to me and Yazeed bin Mu'awiyah. Greetings.""
http://www.maaref-foundation.com/english/library/pro_ahl/imam03_husain/nafasul_mahmoom/22.htm (p. 196)
Other books of reference with identical accounts would be Maqtal al-Husain by al-Muqarram, Maqtal al-Husayn by Abu Mikhnaf, Kitab al-Irshad of al-Mufid, al-Luhuf fi Qatla 'l-Tufuf by Ibn Tawus among others.

Second Answer :

He was prevented by Hurr ibn Yazid Riahi and Umar ibn Sa'ad to leave there without accepting allegiance of Yazid ibn Muawiya. Quoting the wikipedia page of Hurr ibn Yazid Riahi: After the afternoon prayer on the same day, al-Husayn calls his followers to prepare for departure. Al-Hurr swiftly prevents al-Husayn from retracting his steps towards Mak’kah. This incurs al-Husayn to deliver a speech entailing all present to fear God, recognize the rights of all people upon each other, the rights of Prophet Muhammad’s family. He also mentions that if neither al-Hurr nor his soldiers understand these rights, he simply desires to leave in peace. Al-Hurr halts and reconsiders his stance. When the battle started in Ashura (10th of Muharram, 61 AH), Al-Hurr decided to leave the army of Yazid and joined the army of Al-Hussain, the offspring of prophet Muhammad (PBUH): Immediately after Umar ibn Sa'd's verdict to wage battle, al-Hurr approaches al-Husayn with 30 of his horsemen. He expresses his major transgressions to al-Husayn by preventing access back to Medina, and compelling them to halt at Karbala. By omitting himself from al-Husayn’s opposition in the final moments of the Battle at Karbala, al-Hurr’s death embeds itself within the final martyrdom of al-Husayn. It is never seen that a military commander joins the opposite army during the battle - while he is sure he will be sacrificed as there were only 72 people in front of thousands - and this shows which army was in the correct way.

Third Answer :

1. Actually Imam didn't leave Mecca to save his life, but to respect Mecca as he knew Yazid will not respect the city and Al-Haraam Mosque (as he didn't and fight against the city some years later in a war with Abdullah bin Zubair). Imam already knew he will be killed and even he knew that he was to be killed in Karbala. Even the name of Karbala was Karbala from decades or centuries before due to the event that had not been happened before Imam Hussain's battle in Karbala, since Karbala is a shorten form of "Karb o Bala" (کرب و بلا) which points to what happened for Imam and his family and friends during half a day.
2. Imam Hussain had the science of his grand father the holy prophet the city of knowledge, and his father Imam Ali the successor of the holy prophet and the gate to that city of knowledge, with him, peace be upon all of them. He could if he wished to ask Allah kill all of those who were to kill him but he said we are those about whom people are examined. And people were examined in Karbala, some like Hur ibn Yazid Ar-Riyaahi passed the exam successfully and some like Yazid, ibn Sa`d and Ubeidullah ibn Ziyad failed. See how Allah says about Yazid in this regard:

وَإِذْ قُلْنَا لَكَ إِنَّ رَبَّكَ أَحَاطَ بِالنَّاسِ ۚ وَمَا جَعَلْنَا الرُّؤْيَا الَّتِي أَرَيْنَاكَ إِلَّا فِتْنَةً لِّلنَّاسِ وَالشَّجَرَةَ الْمَلْعُونَةَ فِي الْقُرْآنِ ۚ وَنُخَوِّفُهُمْ فَمَا يَزِيدُهُمْ إِلَّا طُغْيَانًا كَبِيرًا

Behold! We told thee that thy Lord doth encompass mankind round about: We granted the vision which We showed thee, but as a trial for men,- as also the Cursed Tree (mentioned) in the Qur'an: We put terror (and warning) into them, but it only increases their inordinate transgression! [or more precisely translating the last part of the verse: their Yazid was nothing but an inordinate transgression] [17:60]