There are many intellectual and traditional reasons for the Wilaya and leadership of the Ahl al-Bayt (a). From the viewpoint of intellect and according to the philosophy of Imamate, infallibility is one of the most important requirements of an Imam (a). Verses from the Qur'an also support this issue (Cf. the verse of Uli al-Amr and the verse of Sadiqin).
On the other hand, Ahl al-Bayt (a) are infallible, thus imamate and leadership of the Islamic Ummah after the Prophet (s) is upon them. Moreover, obeying the Ahl al-Bayt (a) is obligatory. The reasons of the obligation of following the Ahl al-Bayt (a) are absolute and includes all obligations and prohibitions of the lives of Muslims and does not make any difference between worshipping, economic, political and cultural issues. For example in the verse of Uli al-Amr, obeying the Uli al-Amr has the very same position and wide usage which obeying the Prophet (s) has. As it was explained, Uli al-Amr are infallible as this quality only belongs to the Ahl al-Bayt (a) of the Prophet (s).
From the discussions about the love of the Ahl al-Bayt (a), it can be understood that such a love has an introductory role and the ultimate goal is that people find the path of truth and by following it, they reach happiness. Finding and walking the path of truth includes all individual, social, worship and political aspects of life.
The important point in this regard is that, in many reports of the hadith of Thaqalayn, both adherence to the Ahl al-Bayt (a) and the Wilaya of Imam 'Ali (a) have been mentioned. In other words, in the occasion of Ghadir Khumm, the Prophet (s) talked both about his Ahl a-Bayt (a) and advised Muslims to follow them and introduced Imam 'Ali (a) as the Wali and leader of the Islamic Umma after himself. Doing so, the Prophet (s) showed that Wilaya and leadership of Imam 'Ali (a) is the first step for actualizing the message of hadith of Thaqalayn.
Another point is that in some reports of the hadith of Thaqalayn, the Qur'an and the Ahl al-Bayt (a) have been referred to as Khalifatayn, "Indeed, I have left among you two caliphs: the Book of God and my Ahl al-Bayt (a). According to this hadith, Ahl al-Bayt (a) are successors of the Prophet (s) and their rule is overall and encompassing.
In another hadith, after mentioning that the Prophet (s) introduced the Qur'an and the Ahl al-Bayt (a) as the two important legacies he left for the Islamic Umma, it is mentioned that, "the earth will never be empty of the Ahl al-Bayt (a), otherwise, it will become furious over its dwellers. Then he (s) said, " 'O God! You will never leave the earth empty from Your Hujja [proof]. They are few in numbers and great in position before You." Therefore, Ahl al-Bayt (a) are the Hujja [proofs] of God on earth, and imamate and leadership belongs them.
Another evidence for the fact that hadith of Thaqalayn implies the imamate of the Ahl al-Bayt (a) is that Imam 'Ali (a) has referred to it as proof and evidence in different occasions, one of which is was on the Day of Shawra.
Another evidence is referring to this hadith when debating with Talha, 'Abd al-Rahman b. 'Awf and Sa'd b. Waqqas. Another evidence is referring to this hadith at the time of the caliphate of 'Uthman in the mosque of the Prophet (s) in front of a gathering of companions of the Prophet (s).
Ahmad b. Hanbal narrated from Abu Hurayra that the Prophet (s) looked at 'Ali (a), Fatima (a), Hasan (a) and Husayn (a) and said, "I am at war with anyone who wars with you and at peace with anyone who is at peace with you." From this narration, it can be understood that obeying them is obligatory.