Allah will manifest Himself to His servants in the Hereafter in general, and to Abu Bakr (may Allah be pleased with him) in particular

It is narrated from Sayyiduna Jabir bin Abdullah (may Allah be pleased with them both); he said: We were sitting with the Prophet (peace and blessings be upon him) when the delegation of Abd al-Qays came to him (peace and blessings be upon him). Some of them said things that were idle and pointless. the Prophet (peace and blessings be upon him) turned to Abu Bakr (may Allah be pleased with him) and said: "O Abu Bakr, did you hear what they said?" He said: "Yes, O Messenger of Allah, and I have understood it too." He (peace and blessings be upon him) said: "Then answer them." The narrator said: Abu Bakr answered them, and gave an excellent answer. Then the Messenger of Allah (peace and blessings be upon him) said: "O Abu Bakr, Allah has granted you al-Ridwan al-Akbar (the greatest good pleasure)." One of the people said: "O Messenger of Allah, what is al-Ridwan al-Akbar?" He (peace and blessings be upon him) said: "In the Hereafter, Allah Almighty will manifest Himself to all His servants in general, but He will manifest Himself to Abu Bakr in particular."

It is narrated from Sayyidah Aishah (may Allah be pleased with her) that if the Messenger of Allah (peace and blessings be upon him) had been going to appoint a successor, he would have appointed Abu Bakr and Umar (may Allah be pleased with them both) as successors.
This hadith is sahih (authentic) on the condition of the two Shaykhs (Bukhari and Muslim), but they did not record it!

It is narrated from Sayyiduna Abdullah bin Mas'ud (may Allah be pleased with him) that whatever the Muslims consider good is good with Allah as well, and whatever they consider bad is bad with Allah as well; and it was the unanimous view of all his (peace and blessings be upon him) Companions that we should make Abu Bakr (may Allah be pleased with him) the caliph.
This hadith is sahih (authentic) in its chain of narration, but the two Shaykhs (Bukhari and Muslim) did not record it. It has a supporting narration (shahid) that is more authentic than it, except that there is irsal (a missing link in the chain) in it.

It is narrated from Sayyiduna Abdullah bin Mas'ud (may Allah be pleased with him); he said: When the Prophet (peace and blessings be upon him) passed away, the Muhajirun and the Ansar gathered at the Saqifah of Banu Sa'idah to pledge allegiance to Abu Bakr (may Allah be pleased with him). So I went to the Mother of the Believers, Sayyidah Umm Salamah (may Allah be pleased with her), and told her that the people had pledged allegiance to Abu Bakr (may Allah be pleased with him).

It is narrated from Abu Wa'il that it was said to Sayyiduna Ali bin Abi Talib (may Allah be pleased with him): "Will you not appoint a successor (khalifah)?" He said: "the Messenger of Allah (peace and blessings be upon him) did not appoint a successor, that I should do so. However, if Allah Almighty intends good for the people, He will unite them after me upon the best of them, just as He united them after their Prophet (peace and blessings be upon him) upon the best of them."
This hadith is sahih (authentic) in its chain of narration, but the two Shaykhs (Bukhari and Muslim) did not record it. (Mention of the authentic narrations from the Companions (may Allah be pleased with them) that state their consensus on the caliphate of Sayyiduna Abu Bakr al-Siddiq (may Allah be pleased with him) and their addressing him as "Ya Khalifata Rasulillah" (O Successor of the Messenger of Allah).)