In a Nutshell:
The Prophet (saw) when praying in congregation asked we stand side by side in a straight line and not leave any gaps. People do not need to stand too close together whereby they touch or cause discomfort and distraction.
Anas relates the Prophet (saw) having said:
كَانَ رَسُولُ اللَّهِ صَلَّى اللَّهُ عَلَيْهِ وَسَلَّمَ يُقْبِلُ عَلَيْنَا بِوَجْهِهِ، قَبْلَ أَنْ يُكَبِّرَ فَيَقُولُ: "تَرَاصُّوا وَاعْتَدِلُوا،
"The Prophet (saw) would turn his face to us before he began the salah and he (saws) would say: 'Be close together and straighten your rows.'" (Ahmed 12277.)
Anas reports the Prophet (saw) would say:
سَوُّوا صُفُوفَكُمْ فَإِنَّ تَسْوِيَةَ الصُّفُوفِ مِنْ تَمَامِ الصَّلاَةِ
"Make your rows straight for the straightening of the rows is part of the completion of the salah." (Sahih al-Bukhari 723, Sahih Muslim 433.)
An-Nu'man ibn Bashir says:
كَانَ رَسُولُ اللَّهِ صلى الله عليه وسلم يُسَوِّي صُفُوفَنَا حَتَّى كَأَنَّمَا يُسَوِّي بِهَا الْقِدَاحَ حَتَّى رَأَى أَنَّا قَدْ عَقَلْنَا عَنْهُ ثُمَّ خَرَجَ يَوْمًا فَقَامَ حَتَّى كَادَ يُكَبِّرُ فَرَأَى رَجُلاً بَادِيًا صَدْرُهُ مِنَ الصَّفِّ فَقَالَ " عِبَادَ اللَّهِ لَتُسَوُّنَّ صُفُوفَكُمْ أَوْ لَيُخَالِفَنَّ اللَّهُ بَيْنَ وُجُوهِكُمْ" .
"The Prophet (saw) would straighten us in our rows as one straightens an arrow, until he saw that we had learned from him. One day, he saw a person with his chest sticking out and he said: 'You had better straighten your rows or Allah will cause differences among you." (Bukhari 717, Sahih Muslim 436, and Tirmidhi)
Abu Umamah reports that the Prophet (saw) said:
سَوُّوا صُفُوفَكُمْ وَحَاذُوا بَيْنَ مَنَاكِبِكُمْ وَلِينُوا فِي أَيْدِي إِخْوَانِكُمْ ، وَسُدُّوا الْخَلَلَ ؛ فَإِنَّ الشَّيْطَانَ يَدْخُلُ فِيمَا بَيْنَكُمْ بِمَنْزِلَةِ الْحَذْفِ
"Straighten your rows and put your shoulders close to each other and be gentle with each other and fill in the gaps for the Shaytaan passes through what is between you like small sheep [are able to pass through gaps]." (Ahmad 22263, and Tabarani 7727.)
Jabir ibn Samrah says:
خَرَجَ إِلَينَا رَسُولُ اللَّهِ صَلََّى اللهُ عَلَيهِ وَسَلَّمَ فَقَالً " أَلَا تَصُفُّونَ كَمَا تَصُفُّ الْمَلاَئِكَةُ عِنْدَ رَبِّهِمْ ". قَالُوا وَكَيْفَ تَصُفُّ الْمَلاَئِكَةُ عِنْدَ رَبِّهِمْ قَالً " يُتِمُّونَ الصَّفَّ الْأَوَّلَ ثُمَّ يَتَرَاصُّونَ فِي الصَّفِّ ". n't you make the rows like the Angels make their rows in the presence of their Lord?' We asked: 'O Messenger of Allah (saw), how do the Angels make their rows in the presence of their Lord?' He (saw) replied: 'They complete the first row and stand closely together, side by side, in the row.'" (Sunan an-Nasa'i 816.)
Imam Shawkani said:
وَالْمُرَادُ بِتَسْوِيَةِ الصُّفُوفِ : اِعْتِدالُ الْقَائِمِينَ بِهَا عَلَى سَمْتٍ وَاحِد
"The meaning of 'Make your rows straight,' the straightness of prayer's performers in one way." (Nayl al-'Awttar, Vol. 2, p. 199.)
The Shafi'i jurist Al-Nawawi said in Sharh Muslim:
With regards to the phrase "[he] used to straighten our rows, as if he was straightening the shaft of an arrow", the qadaah is the wooden shaft of the arrow where it is pared and trimmed, i.e., he paid such great attention to straightening the rows, as if he was straightening the shaft of an arrow, because they were so straight.
Prayer congregation should be foot to foot, and shoulder to shoulder in a straight row as much as possible. The Imam should organize the first row before praying and worshiper could do that as well with kindness and in a way that does not break the khushu'.
For more elaboration in the subject, kindly read: Are Muslims required to literally stand foot to foot and shoulder to shoulder in prayer?
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