These verses refer to the Banu Qurayzah tribe (5 AH, 627 AD) breaking treaties and siding with enemies against the Muslims (Tanwir al-Miqbas min Tafsir Ibn ‘Abbas and Zamakhshari). We have written about this incident in detail here: ‘Re-Examining Banu Qurayzah Incident‘.
8:56 The ones with whom you made a treaty but then they break their pledge every time, and they do not fear Allah.
8:57 So if you, [O Muhammad], gain dominance over them in war, disperse by [means of] them those behind them that perhaps they will be reminded.
8:58 If you [have reason to] fear from a people betrayal, throw [their treaty] back to them, [putting you] on equal terms. Indeed, Allah does not like traitors.
8:59 And let not those who disbelieve think they will escape. Indeed, they will not cause failure [to Allah].
8:60 And prepare against them whatever you are able of power and of steeds of war by which you may terrify the enemy of Allah and your enemy and others besides them whom you do not know [but] whom Allah knows. And whatever you spend in the cause of Allah will be fully repaid to you, and you will not be wronged.
8:61 And if they incline to peace, then incline to it [also] and rely upon Allah. Indeed, it is He who is the Hearing, the Knowing.
8:56 – The Banu Qurayzah are the ones mentioned here breaking the treaty that they agreed to abide by, that they would not side with enemies against the Muslims. God Almighty says that they do not fear God, and that they break treaties every time (Muqatil ibn Sulayman al-Balkhi – Muqatil Tafsir, Tabari and Tafsir Baghawi).
8:57 – Here the believers are told to inflict on those hostile enemies a punishment, as to stop further blood from being shed.
8:58 – The only time that is acceptable to throw a treaty back is when it is investigated and found with evidence that they have broken the treaty, and are treacherous (Tanwir al-Miqbas min Tafsir Ibn ‘Abbas, Tafsir al-Jalalayn and Razi).
8:59 – Here God states that those who have persecuted, and done horrible things to innocent believers wont be forgotten, and those responsible will be caught eventually (Tanwir al-Miqbas min Tafsir Ibn ‘Abbas). This verse eludes to the Banu Qurayzah and others who wronged the Muslims, 1400 years ago.
8:60 – Those who waged war, there was only one way to stop them from shedding further blood, by the believers to take up arms against the enemies, and put fear in their hearts, 1400 years ago (Tanwir al-Miqbas min Tafsir Ibn ‘Abbas).
8:61 – If the enemy (Banu Qurayzah) offered peace, and stopped their hostilities, the believers were commanded by God to stop fighting them, and make a treaty with them (Tanwir al-Miqbas min Tafsir Ibn ‘Abbas and Tafsir Ibn Kathir). Even when the enemy were hostile, persecuted believers, the Muslims were always to resort to peace if the enemy sought it.
This shows shows that the wars the Muslims were involved in at the time when Muhammed (p) was alive, was always done so as a consequence of the hostilities, and violence enemies committed. Hence, the goal of the Muslims was peace, not war.
Malik Ghulam Farid:
“1134. They repeatedly break their plighted word and dishonour agreements solemnly entered into.
1135. The believers are enjoined never to take up arms without a valid cause. But once they do, they should fight so valiantly and deal such deadly blows as should strike terror in the hearts of their enemies. A feebly pursued and lingering war is never a wise policy. If there is to be fighting it should be swift and deadly.
1136. If a people with whom Muslims have entered into a covenant dishonour it, they should be plainly told that the covenant has come to an end and Muslims, if attacked, would fight back with all the force at their command. But under no circumstances are Muslims permitted to make a surprise attack, without giving prior notice.
Ala sawa’in means, on terms of equality, i.e., in such a manner that each party should know that it is free of its obligations.
1137. Quwwah signifies all the forces at the command of Muslims, including all sorts of implements of war, etc.
1138. For Ribat see 554, 555.
1139. The verse tells Muslims that efficient preparation is the best means of preventing war and enjoins them that they should not only keep sufficient force inside the country but should also station adequate troops on the frontiers and should conduct themselves with such wisdom, faith and energy that the enemy in areas far away from the site of fighting should be so impressed as to give up all idea of fighting them.
The verse also points to the necessity of spending freely in war. It seems to contain also a warning and a prophecy for believers. The Prophecy is that the pagan Arabs are not their only enemies. There are other people who would attack them in the near future. The prophecy referred to the Byzantine and the Persian Empires with whom Muslims had to fight soon after the death of the Holy Prophet.
1140. The verse, besides embodying an important principle about making of peace-treaties, throws interesting light on the character of the wars undertaken by Islam. Muslims did not resort to war to force men to embrace Islam but to establish and maintain peace. If any people after having made war upon Muslims sued for peace the latter enjoined not to reject the offer, even if the enemy might be suing for peace only to deceive them and gain time. This shows to what lengths Islam goes to establish peace among nations.” 
Maulana Muhammad Ali:
“56a. It shows how the opponents of Islam disregarded their responsibility and violated their agreements. The use of the words every time with regard to these violations shows clearly that the Muslims never hesitated in making a new agreement when one was violated, but the disbelievers did not even then respect their agreements; hence, as a last resort, the Muslims were allowed to repudiate unrespected agreements (v. 58).
57a. That is, an exemplary punishment should be inflicted on them, so that a stop might be put to further fighting and bloodshed.
58a. If the other party does not remain faithful to the agreement of peace, the Muslims may also repudiate it. The use of the word fear does not indicate that a mere apprehension, unattended with any action on the part of the other party, is sufficient for repudiation. Read it along with v. 62, and the meaning is clear.
60a. Force (Ar. quwwah) means all those things which are a source of strength, including all kinds of implements of war and other defensive and offensive operations. The Muslims had won a victory at Badr, though they were not even well-equipped and had made no preparation for the war.
But they are told that they must in future keep themselves well prepared and avail themselves of all sources of strength, so that the enemy should by their very preparedness assume a peaceful attitude. It was evident that the weakness of the Muslims was a temptation for their opponents to attack them.
62a. The deceit is in relation to what has been said in the previous verse, the meaning being that if they intend to deceive thee under cloak of peace, even in such a case peace is to be accepted.” 
“56 I.e., withdraw.
57 For an explanation of the wide implications of this statement in the context of the law of cause and effect which God has decreed on His creation (and which is described elsewhere in the Qur’an as sunnat Allah. “The way of God”), see my note on the phrase “God does not change men’s condition unless they change their inner selves” occurring in 13:11.
58 Cf. verse 22 of this surah, where the same epithet is applied to human beings “who do not use their reason”. In the present instance, it should be noted, the particle fa at the beginning of the phrase fa-hum la yu’minun has the meaning of “and therefore” (“and therefore they do not believe”): thus showing that lack of belief in spiritual verities is a consequence of one’s being “bent on denying the truth”.
Expressed in positive terms, this amounts to the statement that belief in any ethical proposition depends on one’s readiness to consider it on its merits and to admit the truth of whatever one’s mind judges to be in conformity with other – empirically or intuitively established – truths. As regards the expression alladhina kafaru, the use of the past tense is meant here, as so often in the Qur’an, to stress the element of intention, and is, therefore, consistently rendered by me – wherever the context warrants it – as “those who are bent on denying the truth” (see also surah 2, note 6).
59 Lit., “every time”. The covenants referred to are agreements between the Muslim community and non-Muslim political groupments. Although this passage is addressed in the first instance, to the Prophet, the “thou” relates here to every follower of the Qur’an and, thus, to the Muslim community of all times. With the above verse, the discourse returns to the subject of war with unbelievers to which most of this surah is devoted.
The reference to the unbelievers’ “breaking their covenants” has two implications: firstly, that the establishment of covenants (i.e., of peaceful relations) with non-Muslims is not only permissible but, in fact desirable (cf. verse 61): and, secondly, that the Muslims may resort to war only if and when the other party is openly hostile to them.
60 Lit., “put to flight, by means of them, those who come after them”; or “terrify through them those who follow them”: i.e., “fight against them and inflict an exemplary punishment on them”.
61 The “reason to fear treachery” must not, of course, be based on mere surmise but on clear, objective evidence (Tabari, Baghawi, Razi; also Manar X, 58).
62 I.e., “renounce the covenant in an equitable manner (‘ald sawa’)”., Tabari explains this sentence thus: “Before making war on them, inform them that because of the clear evidence of their treachery thou hast renounced the treaty which existed between thee and them, so that both thou and they should know that thou art at war with them.”
Baghawi, in his commentary on this verse, gives an almost identical interpretation and adds, “so that they should not be under the false impression that thou hast renounced the treaty after having started the war.” Thus the concluding sentence of this verse – “God does not love the treacherous” – is a warning to the believers as well as to their enemies (Manor X, 58 f.). 63 Lit., “that they have outstripped”.
64 Lit., “tethering of horses” (ribat al-khayl): an expression which signifies “holding in readiness mounted troops at all points open to enemy invasion (thughur)”; hence, tropically, the over-all maintenance of military preparedness.
65 Lit., “God’s enemy and your enemy” – implying that every “enemy of God” (i.e., everyone who deliberately opposes and seeks to undermine the moral laws laid down by God) is, eo ipso, an enemy of those who believe in Him.
66 I.e., of resources, efforts and sacrifice of life.
67 The implication is that “even if they offer peace only with a view to deceiving thee, this [offer of] peace must be accepted, since all judgment [of their intentions] must be based on outward evidence alone” (Razi): in other words, mere suspicion cannot be made an excuse for rejecting an offer of peace.” 
 The Holy Qur’an – Arabic Text With English Translation & Short Commentary By Malik Ghulam Farid, Page 370 – 372
 The Holy Quran Arabic Text with English Translation, Commentary and comprehensive Introduction [Year 2002 Edition] by Maulana Muhammad Ali, page 390 – 391
 The Message of The Quran translated and explained by Muhammad Asad, page 358 – 360 http://www.usc.edu/schools/college/crcc/private/cmje/religious_text/The_Message_of_The_Quran__by_Muhammad_Asad.pdf