They give precedence to the Muhaajireen over the Ansaar, because Allaah says (interpretation of the meaning):
And the foremost to embrace Islam of the Muhaajiroon and the Ansaar and also those who followed them exactly (in Faith). Allaah is well-pleased with them as they are well-pleased with Him. He has prepared for them Gardens under which rivers flow (Paradise), to dwell therein forever. That is the supreme success

[al-Tawbah 9:100.] So the aayah gives precedence to the Muhaajireen over the Ansaar. They believe that Allaah said concerning the people of Badr who numbered three hundred and umpteen Do whatever you like, for I have forgiven you, because the Prophet (peace and blessings of Allaah be upon him) said, Maybe Allah looked at the people of Badr and said, Do whatever you like, for I have forgiven you. (Narrated by al-Bukhaari, 3007; Muslim, 2494 - from the hadeeth of Ali ibn Abi Taalib) They believe that no-one who gave the oath of allegiance (bayah) under the tree will enter Hell, as the Prophet (peace and blessings of Allaah be upon him) said. Rather Allaah is pleased with them and they are pleased with Him, and they numbered more than one thousand and four hundred. Allaah says (interpretation of the meaning):
Indeed, Allaah was pleased with the believers when they gave the Bayah (pledge) to you (O Muhammad) under the tree, He knew what was in their hearts, and He sent down As-Sakeenah (calmness and tranquillity) upon them, and He rewarded them with a near victory[al-Fath 48:18]

And the Prophet (peace and blessings of Allaah be upon him) said: In sha Allaah, no one among the companions of the tree, those who gave their oath of allegiance beneath the tree, will enter Hell. (Narrated by Muslim, 2496). Among those who pledged allegiance beneath the tree were Abu Bakr, Umar, Uthmaan and Ali, may Allaah be pleased with them all.

And they (Ahl al-Sunnah) bear witness that those will enter Paradise who the Messenger of Allaah bore witness that they would enter, such as the ten (who were given the glad tidings of Paradise), and Thaabit ibn Qays ibn Shammaas, and others among the Sahaabah (may Allaah be pleased with them). The Prophet (peace and blessings of Allaah be upon him) said: Abu Bakr will be in Paradise, Umar will be in Paradise, Uthmaan will be in Paradise, Ali will be in Paradise, Talhah will be in Paradise, al-Zubayr will be in Paradise, Abd al-Rahmaan ibn Awf will be in Paradise, Sad will be in Paradise, Saeed will be in Paradise and Abu Ubaydah ibn al-Jarraah will be in Paradise. (Narrated by Abu Dawood, 4649; al-Tirmidhi, 3747; classed as saheeh by al-Albaani).

They affirm that which has been narrated in mutawaatir reports from Ali ibn Abi Taalib (may Allaah be pleased with him) and others, that the best among this ummah after its Prophet is Abu Bakr, then Umar. It was narrated that Muhammad ibn al-Hanafiyyah said: I said to my father (Ali ibn Abi Taalib), Which of the people is the best after the Messenger of Allaah (peace and blessings of Allaah be upon him)? He said, Abu Bakr. I asked, Then who? He said, Umar. I was worried that he would say Uthmaan, so I said, Then is it you? He said, I am just a man among the Muslims. (Narrated by al-Bukhaari, 3671). So they (Ahl al-Sunnah) list Uthmaan as the third and Ali as the fourth, may Allaah be pleased with them.

See al-Waasitiyyah by Shaykh al-Islam Ibn Taymiyah, with commentary.

Secondly: it is part of the belief of Ahl al-Sunnah that the most deserving of the people to succeed the Messenger of Allaah (peace and blessings of Allaah be upon him) was Abu Bakr al-Siddeeq (may Allaah be pleased with him). The evidence for the leadership of Abu Bakr (may Allaah be pleased with him) is as follows:

1 It was narrated from Muhammad ibn Jubayr ibn Mutim that his father said: A woman came to the Prophet (peace and blessings of Allaah be upon him) and he told her to come back later. She said, What if I come back and do not find you? as if she was referring to death. He (peace and blessings of Allaah be upon him) said, If you do not find me, then go to Abu Bakr. (Narrated by al-Bukhaari, 3659).

2 It was narrated that Ibn Masood said: The Messenger of Allaah (peace and blessings of Allaah be upon him) said: Take as your leaders those who come after me, Abu Bakr and Umar. (Narrated by al-Tirmidhi, 3805; classed as saheeh by al-Albaani).

3 It was narrated that Ibn Umar (may Allaah be pleased with them both) said: The Messenger of Allaah (peace and blessings of Allaah be upon him) said: Whilst I was standing by a well, drawing water from it, Abu Bakr and Umar came. Abu Bakr took the bucket and drew one or two buckets, but there was some weakness in his drawing. May Allah forgive him. Then Umar bin Al-Khattaab came and the bucket turned into a very large one in his hands. I had never seen such a mighty person as he in doing such hard work till all the people drank to their satisfaction and watered their camels that knelt down there. (Narrated by al-Bukhaari, 3676).

Ibn Hajar said in his commentary on this hadeeth:

Whilst I was standing by a well means, in a dream. Drawing water from it means, filling a bucket with water. He drew one or two buckets (dhanooban aw dhanoobayn): dhanoob refers to a large bucket when it is full of water. It seems to me that this refers to the major conquests which took place during his reign (as khaleefah), which numbered three. Hence the hadeeth does not refer to the number of buckets in the case of Umar, rather it describes his drawing of water as mighty, referring to the many conquests that would occur during his reign as khaleefah. And Allaah knows best.

Al-Shaafai mentioned the interpretation of this hadeeth in al-Umm, where after quoting it he said: The meaning of the phrase but there was some weakness in his drawing is that his time of leadership was short, because of his untimely death and the fact that the war against the apostates (ahl al-riddah) kept him from reaching the level of conquests attained by Umar during his lengthy reign as khaleefah.

With regard to the phrase May Allaah forgive him, al-Nawawi said: This is a duaa on the part of the one who is speaking, i.e., there can be no other interpretation. Others said that it was an indication that the death of Abu Bakr was close. A similar case is to be seen in the aayah in which Allaah says to His Prophet (peace and blessings of Allaah be upon him):

So glorify the Praises of your Lord, and ask His forgiveness. Verily, He is the One Who accepts the repentance and Who forgives[al-Nasr 110:3 interpretation of the meaning]

This aayah was an indication that the death of the Prophet (peace and blessings of Allaah be upon him) was close. I say: it may be that it was an indication that the small number of conquests during his time was not his fault, because the reason for that was the brevity of his rule, and the meaning of forgiveness is to remove any blame from him.

And the bucket turned into a very large one in his hands means, it became a huge bucket. I had never seen such a mighty person as he means that he achieved the ultimate.

4 It was narrated that Aaishah said: The Messenger of Allaah (peace and blessings of Allaah be upon him) said to me when he was sick, Call Abu Bakr for me, your father and your brother, so that I may dictate a letter. For I am worried lest someone who is ambitious says that he is more entitled to the position of leadership, but Allaah and the believers will not accept anyone other than Abu Bakr. (Narrated by Muslim, 2387).

5 During his final illness, the Prophet (peace and blessings of Allaah be upon him) appointed Abu Bakr al-Siddeeq to lead the Muslims in prayer and he did not accept anyone else to take his place. The fact that he was appointed to take his place in the case of minor leadership (leading the prayers) indicates that he would succeed him in the case of major leadership (as imaam or khaleefah).

And Allaah knows best.