While the Prophet [saw] was being buried, some members of the Ansaar heard that certain members of the Muhajiroon sought to take power, and felt that they would be maligned and put to one side and so decided to preemptively select someone among themselves. Umar ibn Al Khattab narrates the event as chaotic and a catastrophe the Ummah in his eyes, had been saved from. In Saheeh-Al-Bukhari:
“And no doubt after the death of the Prophet (ﷺ) we were informed that the Ansar disagreed with us and gathered in the shed of Bani Sa`da. `Ali and Zubair and whoever was with them, opposed us, while the emigrants gathered with Abu Bakr. I said to Abu Bakr, ‘Let’s go to these Ansari brothers of ours.’ So we set out seeking them, and when we approached them, two pious men of theirs met us and informed us of the final decision of the Ansar, and said, ‘O group of Muhajirin (emigrants) ! Where are you going?’ We replied, ‘We are going to these Ansari brothers of ours.’ They said to us, ‘You shouldn’t go near them. Carry out whatever we have already decided.‘ I said, ‘By Allah, we will go to them.’ And so we proceeded until we reached them at the shed of Bani Sa`da. Behold! There was a man sitting amongst them and wrapped in something. I asked, ‘Who is that man?’ They said, ‘He is Sa`d bin ‘Ubada.’ I asked, ‘What is wrong with him?’ They said, ‘He is sick.’ After we sat for a while, the Ansar’s speaker said, ‘None has the right to be worshipped but Allah,’ and praising Allah as He deserved, he added, ‘To proceed, we are Allah’s Ansar (helpers) and the majority of the Muslim army, while you, the emigrants, are a small group and some people among you came with the intention of preventing us from practicing this matter (of caliphate) and depriving us of it.’
There didn’t seem to be any sort of understanding from some members of the Ansaar who had gathered there that Abu Bakr ibn Quhafa, or Umar ibn Al Khattab, or Uthman ibn Affan or Abu Ubaidah were superior than most of the muslims. Rather it was a power struggle, and the few members of the Ansaar at Saqifah had accused Abu Bakr, and Umar ibn Al Khattab and a few emigrants of wanting to take power from them. In fact, Umar ibn Al Khattab only goes to Saqifah because he is told that some members of the Ansaar have already pre-emptively decided a leader amongst them.
Interesting to note – and this will be covered in the next section is that Ali ibn Abi Talib and others opposed Abu Bakr and Umar. They had heard about what they were doing with regards to leadership and boycotted them and gathered in the house of Fatima binte Muhammed [saw]. Some members of the Ansaar however, decided to choose their own leader before Abu Bakr or Umar or some of the emigrants could influence or say anything.
In the same Narration of Saheeh-Al-Bukhari, Umar ibn Al-Khattab continues after Abu Bakr had rose up to ask the Ansaar to nominate either Umar or Abu Ubaidah as the caliphs:
“..And then one of the Ansar said, ‘I am the pillar on which the camel with a skin disease (eczema) rubs itself to satisfy the itching (i.e., I am a noble), and I am as a high class palm tree! O Quraish. There should be one ruler from us and one from you.’ Then there was a hue and cry among the gathering and their voices rose so that I was afraid there might be great disagreement, so I said, ‘O Abu Bakr! Hold your hand out.’ He held his hand out and I pledged allegiance to him, and then all the emigrants gave the Pledge of allegiance and so did the Ansar afterwards. And so we became victorious over Sa`d bin Ubada (whom Al-Ansar wanted to make a ruler).”
It is very interesting to note that even after Abu Bakr speaks, the Ansaar still do not want to elect him, and there is an enormous quarrel and in-fighting, so much so voices began to rise to such a level Umar ibn Al Khattab feared there would ensue chaotic scenes. It is here where Umar ibn Al Khattab does not really elaborate further, as he just mentions on impulse getting Abu Bakr to put his hand out so he and the few emigrants with him can give allegiance to him. What then led some of the Ansaar to go ahead with this ‘afterwards’ is up to debate and question. Some historians have noted the rivalries between the Aws and Khazraj resurfacing.
Suffice to say that what occurred was only to be expected – a chaotic power struggle for the coveted authority and rule among the muslims the Prophet [saw] had. As Umar ibn Al Khattab concludes in the same tradition
“`Umar added, “By Allah, apart from the great tragedy that had happened to us (i.e. the death of the Prophet), there was no greater problem than the allegiance pledged to Abu Bakr because we were afraid that if we left the people, they might give the Pledge of allegiance after us to one of their men, in which case we would have given them our consent for something against our real wish, or would have opposed them and caused great trouble.”
Interestingly, we find that many of the Ansaar would have chosen Ali ibn Abi Talib, however, what had occurred was that they heard plans by a small number of the emigrants who sought to secure power. Ali ibn abi Talib, members of the Banu Hashim, and a number of the companions opposed Abu Bakr and Umar. However, some of the other Ansaar saw that while some of the emigrants had sought power for themselves, rather than being maligned and pushed to the sidelines and have others rule over them who had disobeyed the command of the Prophet [saw], it was better for them to pre-empt this and choose a leader for themselves.
We again will present the traditions from Tarikh-Tabari, where some among the Ansaar state that they would only choose Ali ibn Abi Talib [when faced with the opposition of Abu Bakr, Umar, and Abu Ubaidah].
” We do not deny the merits of those you have mentioned. Indeed there is among you a person with whom if he seeks authority, none will dispute [i.e. Ali]‘”.
Reference:  Tarikh, by al Yaqubi, Volume 2 page 113-114, quoted from History of Tabari, Volume 9 English translation.
Just to note, Abu Bakr, Umar, and Abu Ubaydah were already present at Saqifah when the scuffle was taking place. The greatest man on that list according to sunni’s by consensus [as Uthman was not mentioned] was Ali ibn Abi Talib [as]. It could only have been him the was meant, even by logic of our sunni brothers/sisters in islam. This is evident given he was not present, and they had said that one of the men in the list they would not have disputed and if they wanted authority , they would not dispute it.
[Tarikh -At Tabari[: Ibn Humayd-Jarir – Abu Ma’shar Ziyad b. Mughzrahl-Kulayb-Abu Ayyub-Ibrahim- “The Ansar gathered in a roofed building (saqifah) of the Banu Sa’idah to render their oath of allegiance to Sa’d b. Ubadah. This news reached Abu Bakr, so he came to them with’Umar and Abu’Ubaydah b. al-Jarrah, asking [them] why [they had gathered]. They replied, “Let us have a ruler (amir) from us and another from you.” Abu Bakr said, “The rulers will be from us, and the viziers from you.” Abu Bakr then added, “I am pleased [to offer] you one of these two men: ‘Umar or Abu ‘Ubaydah. Some people came to the Prophet asking him to send a trustworthy man with them. The Messenger of God said that he would send a truly trustworthy man with them, and he sent Abu ‘Ubaydah b. al-Jarrah. I am pleased [to offer] you Abu ‘Ubaydah.” ‘Umar stood up saying, “Who among you would be agreeable to leave Abu Bakr whom the Prophet gave precedence? and he gave him the oath of allegiance. The people followed [‘Umar]. The Ansar said, or some of them said, “We will not give the oath of allegiance [to anyone] except ‘Ali.”
Reference: History by Tabari.
It is worthy to note that Ali ibn Abi Talib, members of the Banu Hashim and many companions did not give their oath of allegiance to Abu Bakr. It was a chaotic power struggle, and then he became the Caliph, and anyone who disobeyed the orders of giving their allegiance were ordered to give it, lest they be committing treason. Abu Bakr, Umar, and others were senior companions and commanded fear and respect, and it was time of confusion and Fitnah for the Ummah.